Structure and Decay in the QED Vacuum

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Authors Labun, Lance

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Link to Item http://hdl.handle.net/10150/203491 STRUCTURE AND DECAY IN THE QED VACUUM

Lance Andrew Labun

A Dissertation Submitted to the Faculty of the

DEPARTMENT OF PHYSICS

In Partial Fulﬁllment of the Requirements For the Degree of

DOCTOR OF PHILOSOPHY

In the Graduate College

THE UNIVERSITY OF ARIZONA

2011 2

THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE

As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Lance Andrew Labun entitled Structure and Decay in the QED Vacuum and recommend that it be accepted as fulﬁlling the dissertation requirement for the Degree of Doctor of Philosophy.

Date: 16 November 2011 Johann Rafelski

Date: 16 November 2011 Sumitendra Mazumdar

Date: 16 November 2011 Michael Shupe

Date: 16 November 2011 Shufang Su

Date: 16 November 2011 Ubirajara van Kolck

Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the ﬁnal copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulﬁlling the dissertation requirement.

Date: 16 November 2011 Dissertation Director: Johann Rafelski 3

STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulﬁllment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED: Lance Andrew Labun 4

ACKNOWLEDGMENTS

Prof. Johann Rafelski, my mentor, has been a continuous source of ideas, energy and wisdom. The deep and broad foundation I now have in the scholarly pursuit of physics is a credit to his teaching.

I am indebted to Prof. Keith Dienes for bringing me to Arizona and providing early guidance.

I would like to thank Prof. Berndt Mueller for encouragement and insightful discussion over the past four years.

I am very grateful to my dissertation committee who granted patience, understanding and helpful advice in the completion of this dissertation.

Many more have supported and contributed to the work constituting this dissertation. The limits of this page leave insuﬃcient space to individually thank each professor, colleague and friend who has enriched my education over the course of the past four years, though even the smallest element is essential to the ﬁnal composition presented here. 5

DEDICATION

I dedicate this work to my parents, Lance C. Labun, Ph.D, and Patricia A. Labun, Ph.D. 6

TABLE OF CONTENTS

LIST OF FIGURES...... 8

ABSTRACT...... 9

CHAPTER 1 Introduction ...... 11 1.1 Strong Field QED: New Tools and Observations...... 11 1.2 Units and Notation...... 13 1.3 External Fields and Vacuum Structure...... 13 1.4 The Euler-Heisenberg Eﬀective Potential...... 15 1.5 Instability and Decay of the Electromagnetic Field...... 18 1.5.1 Vacuum Expectation of the Current...... 19 1.5.2 Imaginary Part of Veﬀ ...... 21 1.5.3 Decay of the Electric Field by Tunneling...... 22

CHAPTER 2 Key Findings ...... 26 2.1 Energy-Momentum Tensor of Nonlinear Electromagnetism...... 26 2.1.1 4-Momentum and Mass...... 27 2.1.2 Rest Frame of an Electromagnetic Field...... 29 2.2 Time Constant for Decay of Electromagnetic Field Mass...... 30 2.3 Energy of Particles from Laser-Induced Vacuum Decay...... 31 2.4 The Energy-momentum Trace...... 34 κ 2.4.1 Fermi Condensate in Vacuum and Tκ ...... 36 κ 2.4.2 Astrophysical Effects of Tκ ...... 37 2.4.3 Modified Lorentz Force...... 38 2.5 Comparison of Modifications of the Maxwell T µν ...... 40 2.6 Dark Energy due to a Metastable Vacuum State...... 42 2.7 Summary and Conclusions...... 43

REFERENCES...... 45

APPENDIX A Energy of the Dirac Vacuum...... 48

APPENDIX B Vacuum Decay Time in Strong External Fields...... 50

APPENDIX C Dark Energy Simulacrum in Nonlinear Electrodynamics... 55

APPENDIX D QED Energy-momentum Trace as a Force in Astrophysics.. 72 7

TABLE OF CONTENTS – Continued

APPENDIX E Strong Field Physics: Probing Critical Acceleration and Inertia with Laser Pulses and Quark-Gluon Plasma...... 79

APPENDIX F Vacuum Structure and Dark Energy...... 104

APPENDIX G Spectra of Particles from Laser-Induced Vacuum Decay.... 110 8

LIST OF FIGURES

1.1 Photon-photon scattering...... 11 1.2 Diagrammatic representation of the eﬀective potential...... 16 1.3 Example diagrams not included in the eﬀective potential...... 18 9

ABSTRACT

This thesis is a guide to a selection of the author’s published work that connect and contribute to understanding the vacuum of quantum electrodynamics in strong, prescribed electromagnetic fields. This theme is elaborated over the course of two chapters: The first chapter sets the context, defining the relevant objects and conditions of the study and reviewing established knowledge upon which this study builds. The second chapter organizes and explains important results appearing in the published work. The papers

1.(Labun and Rafelski, 2009) “Vacuum Decay Time in Strong External Fields”

2.(Labun and Rafelski, 2010a) “Dark Energy Simulacrum in Nonlinear Electro- dynamics”

3.(Labun and Rafelski, 2010b) “QED Energy-Momentum Trace as a Force in Astrophysics”

4.(Labun and Rafelski, 2010c) “Strong Field Physics: Probing Critical Acceler- ation and Inertia with Laser Pulses and Quark-Gluon Plasma”

5.(Labun and Rafelski, 2010d) “Vacuum Structure and Dark Energy”

6.(Labun and Rafelski, 2011) “Spectra of Particles from Laser-Induced Vacuum Decay” are presented in their published format as appendices. Related literature is cited throughout the body where it directly supports the content of this overview; more extensive references are found within the attached papers. This study begins with the ﬁrst non-perturbative result in quantum electrodynamics, a result obtained by Heisenberg and Euler(1936) for the energy of a 10

zero-particle state in a prescribed, long-wavelength electromagnetic field. The resulting Euler-Heisenberg effective potential generates a nonlinear theory of electromagnetism and exhibits the ability of the electrical fields to decay into electron- positron pairs. Context for phenomena arising from the Euler-Heisenberg effective potential is established by considering the energy-momentum tensor of a general nonlinear electromagnetic theory. The mass of a field configuration is defined, and I discuss two of its consequences pertinent to efforts to observe vacuum decay. I develop a method for non-perturbative evaluation of a trace component of the energy- momentum tensor and discuss its significance and consequences. I study the effect of the energy-momentum trace as part of a Euler-Heisenberg-generated modification to the Lorentz force. Modifications of the energy-momentum tensor from the Maxwell theory are evaluated numerically and compared to those arising from Born-Infeld electromagnetism and the Euler-Heisenberg effective potential for a scalar electron. Finally, I explore how this study guides investigation into how vacuum structure can generate the cosmological dark energy. 11

CHAPTER 1

Introduction

1.1 Strong Field QED: New Tools and Observations

Quantum electrodynamics (QED) exhibits rich dynamics even in the state empty of real electrons or positrons, the vacuum. An electromagnetic field excites virtual electron-positron pairs, and this polarization of the vacuum leads to an effective interaction permitting the scattering of photons. Shown diagrammatically in Fig. 1.1 for two incident photons, the interaction potential was computed in the low photon- energy limit by Euler and Kockel(1935). In electric fields where the potential difference exceeds the rest mass of a pair 2mc2, the electron and positron can become real particles, as recognized by Sauter(1931).

photon e−

photon e+

Figure 1.1: Two-photon scattering mediated by a virtual electron-positron pair ﬂuctuation in the vacuum.

The photon scattering and pair-creation processes are complementary aspects of polarization of the electron-positron pair fluctuations by the external electromagnetic field. Considering the exactly-solvable case of a constant prescribed field, Heisenberg and Euler(1936) unified these aspects in a non-perturbative effective potential for the electromagnetic field, which incorporates all quantum corrections due to one-(electron-positron)-loop. 12

Consequences of the electromagnetic ﬁeld-induced polarization of the vacuum are challenging to observe because they are controlled by the ﬁeld scale

2 2 ~ m c 9 ~ ~ 18 |Bc| = = 4.41 10 T, |Ec| = c|Bc| = 1.32 10 V/m. (1.1) e~ For comparison, a static magnetic field strong by laboratory standards is 1 − 5 T. However in the past two decades, astronomers have begun to identify signatures of astrophysical environments where magnetic fields at the scale Eq. (1.1) may be present. Detailed observations of pulsars have indicated the presence of dipole magnetic fields of 108 − 109 T, and other compact stellar objects are suggested to have fields in excess of 1010 T. See Lattimer and Prakash(2007) and references therein. In recent years, advances in high-intensity laser technology have brought prescribed field-induced polarization effects closer to experimental realization. Tech- niques to compress and shape a laser pulse can concentrate the pulse’s energy into a volume comparable to the laser wavelength, creating ultra-high intensity fields at the focus (Mourou et al., 2006). The critical field Eq. (1.1) expressed as an intensity is ~ 2 29 2 Ic = c0|Ec| = 4.65 10 W/cm (1.2)

−12 (0 = 8.854 10 F/m is the permittivity of the vacuum.) Optical lasers (~ω ' 1 − 5 eV) now routinely achieve intensities in excess of 1020 W/cm2 (Batha et al., 2008). Facilities in planning and construction stages will extend the reach in intensity to 1024 W/cm2 (Korn and Antici, 2009). Much recent work has explored ways to utilize these new experimental capabilities toward observing effects of the field-structured vacuum. These newly recognized strong electromagnetic field environments motivate specific questions posed in the investigation here; they guide the selection and construction of examples in specific evaluations. Yet the lessons and insights from studying QED in strong fields and the long-wavelength effective potential are applicable more widely. For example, this study provides a view of an excited vacuum state that 13

indicates new directions in the eﬀort to understand the cosmological dark energy. Before describing these results and their implications, I use the remainder of this chapter to review the foundations on which this study builds.

1.2 Units and Notation

I use natural units in which ~ = c = 1. The magnitude of the electron charge is √ e = |e| = 4πα. The QED coupling α = (137.036...)−1. With this relationship between e and α, the prefactor (hc)−1 in (Heisenberg and Euler, 1936) becomes (8π2)−1 here (Eq. (1.9) below). A bare lower case m denotes the electron mass m = 0.511 MeV/c2 unless expressly stated otherwise.

The electron field is a four-component spinor, ψa where a = 1, 2, 3, 4 is the Dirac index, and observables such as the expectation value of the current operator are † bilinear in ψa and its Hermitian conjugate ψb . The 4 × 4 Dirac matrices are denoted µ by γab, though the spinor indices will occasionally be suppressed following standard notation (Greiner et al., 1985; Peskin and Schroeder, 1995). Repeated indices are summed over. In covariant expressions, time and space coordinate labels are lower case Greek indices generally chosen from the middle of the alphabet κ, λ, µ, ν... ∈ 0, 1, 2, 3. Lower case Latin indices are used to indicate restriction to spatial coordinate labels i, j, k, ... ∈ {1, 2, 3}. Tensors are contracted with the metric of flat Minkowski space gµν = diag(1, −1, −1, −1). With this choice of metric signature, spacelike 4-vectors have negative squared magnitude 2 µ ν µ a ≡ a a gµν = a aµ < 0, and timelike 4-vectors have positive squared magnitude a2 > 0. When 4-vectors are written out, the 0- (time-)component appears first: aµ = (a0, a1, a2, a3).

1.3 External Fields and Vacuum Structure

Heisenberg and Euler(1936) considered the eﬀect of a long-wavelength (quasi- 14

constant) −1 λ λ¯e = 1/m = (0.511 MeV c/~) (1.3) electromagnetic field on the state containing zero electrons and zero positrons. Throughout this work, this state, the vacuum of the Dirac electron, will be referred to simply as the ‘vacuum,’ despite the presence also of the prescribed (external) electromagnetic field. Explicit knowledge of the single electron energy levels in a constant field permits exact summation of the energy of the polarized states. The calculation is readily demonstrated for |E~ | → 0. Choosing the magnetic field to be aligned withz ˆ-axis, B~ = |B~ |zˆ, the energy levels of the electron are quantized Landau levels,

2 2 2 ~ s,l = pz + m + e|B|(l + 1/2 + s), s = ±1/2, l = 0, 1, 2, ... (1.4)

pz is the electron momentum in thez ˆ-direction and remains a continuous parameter also in the presence of the magnetic ﬁeld. s is the spin, and the transverse momentum ~ ~p⊥ = (px, py) is quantized into the Landau orbits e|B|l. Requiring the radius of the orbit to be small enough to remain within the normalization volume (the two- dimensional area L2), the integral over transverse momentum states is

Z d2p e|B~ | ⊥ → . (2π)2 2π The relativistic dispersion relation means that energy levels are present in positive and negative energy pairs. Both positive and negative energy levels contribute equally to the inﬁnite constant obtained from the vacuum expectation value of the Dirac Hamiltonian   1 X X U = hHˆ i = − − . (1.5) sea 2  s,l s,l s,l>0 s,l<0 h...i denotes the expectation value in the with-ﬁeld vacuum. (See AppendixA for construction of the Hamiltonian Hˆ and discussion of this formula.) Here, the pairing 15

of positive and negative levels makes the sum equivalent to a sum over all states

with s,l < 0,

e|B~ | Z ∞ dp q ~ X X z 2 2 ~ Usea(|B|) = s,l = − pz + m + e|B|(l + 1/2 + s). (1.6) 2π −∞ 2π s,l<0 n,s

Although the sum is divergent, techniques such as ζ-function regularization (Hawk- ing, 1977) can be used to control and subtract the infinite contributions. ~ The finite part of Usea(|B|) is the effective potential Veff for the presence of a quasi-constant electromagnetic field in the vacuum of the Dirac electron. Veff is extracted as the energy shift from the vacuum without the field present

~ (1) ~ Veﬀ = Usea(|B|) − Usea(0) − Veﬀ (|B|). (1.7)

As indicated, this subtraction involves two infinite contributions: Usea(0), a field- (1) ~ ~ 2 independent constant, and Veff (|B|) proportional to |B| . As is now well-understood (1) ~ in context of charge renormalization, Veff (|B|) can be absorbed into the definitions of the field and charge e.

1.4 The Euler-Heisenberg Eﬀective Potential

Translational symmetry of the (quasi-) constant ﬁeld implies the eﬀective potential

Veff can depend only on 1 1 1 S = F µνF = (B~ 2 − E~ 2), P = ε F κλF µν = −E~ · B,~ (1.8) 4 µν 2 4 κλµν the two Lorentz invariants quadratic in the field strength tensor. Only even powers of the field strength tensor are permissible, because Veff must be even in the charge e according to charge conjugation symmetry. Consequently, in the diagrammatic representation of Veff , Fig. 1.2, there appear only even numbers of legs attached to the external field. Quartic and higher Lorentz invariant combinations of the κ λ µ ν field tensor (e.g. Fλ Fµ Fν Fκ ) can be decomposed into combinations of S, P, seeing 16

that there are only three scalar combinations of the 3-vector electric and magnetic ﬁelds, E~ 2, B~ 2, and E~ · B~ . Invariance of QED interactions under transformations simultaneously changing charge and parity requires the pseudoscalar invariant P to appear in even powers. The Euler-Heisenberg eﬀective potential

Z ∞ 2 −1 du −m2u eau ebu u 2 2 2 Veff = 2 3 e − 1 − e (b − a ) (1.9) 8π 0 u tan(eau) tanh(ebu) 3 is a complex function of E,~ B~ , non-perturbative in the strength of the prescribed fields. Here, S, P are contained in the eigenvalues of the field strength tensor:

 p√ ±a = ± S2 + P2 − S, F µξν = λξµ, λ = (1.10) ν p√ ±ib = ±i S2 + P2 + S.

As will be shown in Sec. 2.1.1, these invariants assume the magnitudes of the electric a → |E~ 0| and magnetic ﬁelds b → |B~ 0| in the rest frame of the electromagnetic ﬁeld.

Veﬀ = + + +...

Figure 1.2: Polarization of the electron-positron pair fluctuations by a prescribed electromagnetic field: an infinite sum of processes involving scattering of (even) numbers of photons mediated by a single electron-positron loop. All processes of this type are incorporated in the effective potential computed by Heisenberg and Euler(1936).

Adding to Veﬀ the Lagrangian of Maxwell’s electromagnetism, 1 L = − F µνF = −S, (1.11) Max 4 µν 17

one obtains the total eﬀective Lagrangian of the electromagnetic ﬁeld

Leff = −S + Veff (1.12) to first order in the QED coupling α and including all orders in the external field.

The real part of Veff generates effective photon-photon interactions mediated by quantum fluctuations, shown diagrammatically in Figure 1.2. Due to Veff , the ~ ~ effective theory of electromagnetism encoded in Leff is nonlinear in E, B, the fields associated with the kinetic part of the Lagrangian. Thus the superposition principle of Maxwell’s electromagnetism is violated. Instead, superposition applies to the displacement field tensor ∂L Kµν = − eff −→ F µν (1.13) ∂Fµν Leff →−S which incorporates the displacement fields D,~ H~ in the same format as the field strength tensor F µν. The vacuum becomes in this way analogous to a medium with nonlinear dielectric response. A power series representation of Eq. (1.9) can be constructed in E,~ B~ to exhibit these new field-field interactions. The lowest terms are 2 2α ~ 2 ~ 2 2 ~ ~ 2 Veff ' 4 (E − B ) + 7(E · B) 45m (1.14) 16α3 + (E~ 2 − B~ 2) 2(E~ 2 − B~ 2)2 + 13(E~ · B~ )2 + ... 315m4 the first line being the result of Euler and Kockel(1935). The two terms here, being fourth and sixth order in the fields correspond to the first two diagrams in Fig. 1.2. This series is only semi-convergent (Heisenberg and Euler, 1936; Weisskopf, 1936; Dunne, 2004): the coefficients are proportional to Bernoulli numbers, which diverge factorially. In order to have the correct mass dimension, these effective interaction terms are suppressed by the one scale in the theory, the electron mass. This mass scale translates into the ‘critical’ magnetic and electric field strengths, Eq. (1.1), at which 18

the terms in Eq. (1.14) are of order unity. Approaching this scale, the divergent

series Eq. (1.14) becomes unreliable (Labun and Rafelski, 2010a). Furthermore, Veff is non-perturbative at the origin E,~ B~ → 0, as will be clear below in the discussion of the imaginary part. For this reason, the weak-field series, Eq. (1.14), cannot be considered a perturbative expansion, though I have shown by independent evaluation −3 ~ ~ that it agrees with Eq. (1.9) for fields 10 < |E|/|Ec| < 1. With Eq. (1.9), Heisenberg and Euler(1936) introduced the first non- perturbative calculation in quantum field theory and presaged effective field the-

ory. Terms containing derivatives of the fields could be added to Veff ; however, in quasi-constant prescribed fields they are suppressed by Eq. (1.3). Virtual processes at order α2 and higher have not been incorporated. Figure 1.3 displays diagrams representing α2 processes in the external field.

α2 : ≡ + +...

Figure 1.3: Diagrammatic representation of order α2 processes, which are excluded from the Euler-Heisenberg Veﬀ , Eq. (1.9). The double line in the loop represents the sum of an arbitrary number of interactions with the external ﬁeld, as expanded on the right.

1.5 Instability and Decay of the Electromagnetic Field

The effective potential Eq. (1.9) has an imaginary part, which Heisenberg and Euler (1936) suggest should be interpreted after the manner of resonances in scattering theory and correspond to pair creation. An imaginary part must arise in any nonvanishing electric field due to the (formal) infinite spatial extent of the field: by 19

integrating over a suﬃciently long distance, the potential diﬀerence can always be made to exceed 2m, thus connecting states below the gap in the Dirac spectrum to states above the gap. Schwinger(1951) establishes the relationship between the

imaginary part of Veﬀ , vacuum structure and pair creation by computing the current induced in the vacuum and relating it to the eﬀective potential.

1.5.1 Vacuum Expectation of the Current

The starting point is the Green’s function Z x 0 µ ¯ 0 Gab(x, x ) = ihTb ψa(x) exp ie Aµdx ψb(x )i, (1.15) x0 which is the vacuum expectation of time-ordered field operators and a 4 × 4 matrix in Dirac space. The time-ordering operator Tb is defined by  ¯ 0 0 ψa(x)ψb(x )(x0 > x ) ¯ 0 0 Tb ψa(x)ψb(x ) = (1.16) ¯ 0 0 −ψb(x )ψa(x)(x0 > x0) taking into account the anti-commutation of fermionic field operators. The expo- nentiated line integral in the external potential preserves manifest gauge invariance. Represented here in position space, G is the solution to the Dirac equation

(iD/ − m)G(x, x0) = −δ4(x − x0). (1.17)

Dµ = ∂µ +ieAµ is the covariant derivative, and the slash indicates its scalar product µ with the 4-vector of Dirac matrices: D/ ≡ Dµγ . It will be useful to consider G as an operator deﬁned by G(x, x0) ≡ hx |G | x0i. The operator G satisﬁes

(iD/ − m)G = −I (1.18) where I is the identity operator across spacetime and spinor indices. By implementing explicit charge symmetry, the current operator Z x+ξ µ e ¯ µ µ j (x) = lim ψa(x + ξ), γ ψb(x − ξ) exp ie Aµdx (1.19) µ + ab ξµξ →0 2 x−ξ 20

is related to the Green’s function through a symmetrized time ordering of the ﬁeld operators. The limit incorporates an average over past- and future-pointing timelike vectors ξµ (Johnson, 1965). Taking ξµ → 0 in this way symmetrically within past and future lightcones, the limit of time-ordered operators devolves to

¯ 1 ¯ lim T ψa(x + ξ)ψb(x − ξ) = ψa(x), ψb(x) (1.20) µ + b ξµξ →0 2 using again that Fermi operators anti-commute. Combining Eqs. (1.15), (1.19),(1.20) shows that the vacuum expectation of the current is the trace of the Green’s function

hjµ(x)i = lim ie tr γµG(x + ξ, x − ξ), (1.21) µ + ξµξ →0 tr indicating the sum over spinor indices. Being diagonal also in spacetime indices, the vacuum expectation of the current is diagrammatically represented as a propagator line that is closed on itself, i.e. a loop. µ Now hj (x)i is obtained from the eﬀective action Ieﬀ by variation with respect

to Aµ subject to δAµ(x) vanishing at the boundaries. Therefore, the effective action can be derived in a gauge invariant manner by exhibiting the right side of Z Z 4 4 µ δIeff = d x δVeff = d x δAµ(x)hj (x)i (1.22)

4 0 0 as a total diﬀerential. The operator deﬁned by δAµ(x)δ (x − x ) = hx |δAµ | x i is related to the Green’s function through Eq. (1.18),

µ −1 − eγ δAµ = δ(iD/ − m) = δ(G ). (1.23)

Therefore Eq. (1.22) can be written Z 4 −1 δIeﬀ = d x δ −i tr ln G . (1.24)

The integrand is the eﬀective potential, thus yielding the well-known relation

−1 Veﬀ = −i tr ln G = −i tr ln(iD/ − m) (1.25) 21

The trace can be evaluated by either ζ-function regularization (Hawking, 1977) or the proper time method (Fock, 1937; Schwinger, 1951) to separate the field- independent constant and the charge renormalization term seen in Eq. (1.7): The field independent constant is the 1 subtracted in the integrand of Eq. (1.9). The charge renormalization is the subtracted term quadratic in the fields, 2 2 2 Z ∞ (1) e b − a du −m2u 2α m Veff = 2 e = S ln (1.26) 8π 3 0 u 3π µ where µ is the renormalization scale. Note that the coefficient 2α/3π agrees with the QED β-function evaluated at one loop. Introducing in Eq. (1.25) a complex integral representation of the logarithm, Z ∞ du −1 Veff = i tr exp(−iG u), (1.27) 0 u requires defining the integration contour. The contour is deformed by considering the mass to have an infinitesimal imaginary component m2 → m2 − i with > 0. The imaginary part is then evaluated recalling the identity 1 1 lim = P + iπδ(x) (1.28) →0+ x − i x where P denotes principal value. To compute the trace in this approach, quantum equations of motion in the prescribed electromagnetic field are solved over a fictitious state space in the extra ‘time’ dimension u. The Euler-Heisenberg effective potential can be derived because the equations of motion in a constant field are exactly solvable (Schwinger, 1951). After subtraction and renormalization, the finite result for the effective potential yields Eq. (1.9).

1.5.2 Imaginary Part of Veﬀ

Eq. (1.9) exhibits poles in the integrand giving rise to an imaginary part of Veff . The physical significance of the imaginary part is exhibited by considering the vacuum- to-vacuum amplitude Z iIeff 4 hvac, out | vac, ini = he i, Ieff = d x Leff , (1.29) 22

with Leﬀ given by Eq. (1.12). In quasi-constant ﬁelds, the 4-volume integration can be set to the normalization volume, Z d4x → L3T.

The squared amplitude yields the probability of vacuum persistence,

2 3 Γ = |hvac, out | vac, ini| = exp(−L T 2Im Veﬀ ). (1.30)

Real contributions to the action change only the phase, which becomes unity upon squaring the amplitude, whereas an imaginary contribution results in exponential

decay of the vacuum persistence probability. 2Im Veﬀ is the probability per unit volume per unit time of the decay of the ﬁeld-structured vacuum state. The poles in the integrand of Eq. (1.9) are at eau = nπ for n = 1, 2, ..., so that

2 ∞ e ab X 1 nπb 2 2 Im V = coth e−nπm /ea. (1.31) eﬀ 4π2 n a n=1

Taking the limit |B~ | → 0,

2 ~ 2 ∞ e |E| X 1 2 ~ 2 Im V = e−nπm /e|E| (1.32) eff 4π3 n2 n=1 shows that the presence of the imaginary part arises from the presence of the electric ~ potential. Indeed, Im Veff vanishes in the limit |E| → 0: homogeneous magnetic fields

cannot do work and hence are absolutely stable against pair creation. Im Veff is non- ~ perturbative in e|E|, and even though the zero-field limit is smooth, Im Veff does not exist for e|E~ | → 0. For this reason, the weak-field expansion Eq. (1.14) cannot be a perturbative expansion of Veff around e = 0.

1.5.3 Decay of the Electric Field by Tunneling

Decay of the vacuum means the creation of real electron-positron pairs. This fact is understood in terms of screening of the prescribed ﬁeld by real charges: in suﬃciently 23

strong electromagnetic ﬁelds, the total energy is reduced by creating real pairs of electrons and positrons. However, pair creation is typically suppressed due to the large barrier presented by the energy gap in the electron-positron spectrum,

∆ = 2m. (1.33)

In quasi-constant fields, the absence of dynamics means that quantum tunneling is the only way to cross the barrier. As noted by Casher et al.(1979), the necessity of tunneling leads to the exponential suppression of the imaginary part, Eq. (1.31). For the electron to have significant probability of tunneling, the potential difference must be comparable to the energy gap over the length scale of the wave function. The length scale of the wave function is set by the de Broglie wavelength λ = 1/p. An electron created at threshold has momentum equal to its rest mass p = m, so that the de Broglie

wavelength reduces to the Compton wavelength λ → λ¯e. This instability condition reveals again the critical ﬁeld strength, Eq. (1.1),

d(eU) ∆ 2 dU ~ ~ ∼ = 2m ⇔ − = |E| ∼ |Ec|. (1.34) dx λ¯e dx The tunneling probability can be evaluated in the imaginary time formal- ism (Casher et al., 1979; Labun and Rafelski, 2009). Rotating time by the imaginary unit, t → it, a potential barrier of finite height is inverted into a potential well. The action of a fictitious particle with energy in this formal potential well is the action required for the real particle of energy to tunnel through the real potential barrier. Let us review the computation of the tunneling action for a homogeneous electric field. Thez ˆ-direction is aligned with the field so that the potential U(z) = −e|E~ |z. Since the field is quasi-constant in time, particles are created with zero longitudinal momentum, and the energy of the tunneling state depends only on the momentum perpendicular to the field, 2 2 2 2 = ~p⊥ + m ≡ ⊥. (1.35) 24

The momentum q of the ﬁctitious particle in the potential U(z) obeys the same relativistic dispersion relation as the tunneling state, q 2 2 q = ⊥ − U(z) , (1.36)

and the tunneling action is

Z z0 ~ Itun = q dz, z0 = ⊥/e|E|, (1.37) 0

where z0 is the turning point of the ﬁctitious particle in the harmonic potential 2 ~ 2 U(z) = (e|E|z) . The total action for the pair is twice Itun and exponentiating, one ﬁnds the probability for tunneling ! 2 2 π 2 Γtun(⊥) = | exp(−2Itun)| = exp − ⊥ . (1.38) e|E~ |

The probability of persistence of the vacuum, i.e. no particle creation in the state 2 ⊥, is 1 − Γtun(⊥). The total probability of vacuum persistence is the probability no particles are created in any state labeled by transverse energy T and spin s = ±1/2, ! Y 2 X 2 Γ = [1 − Γtun(⊥)] = exp ln 1 − Γtun(⊥) . (1.39) s,⊥ s,⊥ The density of ﬁnal states is continuous in transverse momentum, and in the quasi- constant approximation the longitudinal momentum is determined by the ﬁeld, Z Z dp dp = dt z = e|E~ |T. z dt P The tunneling action is independent of the spin, and the spin sum s just introduces

a factor 2 distinguished with a subscript s. Setting

~ Z 2 X e|E| d p⊥ = L3T 2 (1.40) s 2π (2π)2 s,⊥ in Eq. (1.39) recovers Eq. (1.30). 25

The imaginary time technique provides a means of directly computing the imaginary part of Veff . It has been extended to include constant magnetic fields and special forms of space and time dependent electric fields (Kim and Page, 2002, 2006, 2007), reproducing Veff obtained by direct integration of the Dirac spectrum (Sauter, 1931) and Green’s function approaches (Nikishov, 1970). Moreover, evaluating the energy levels Eq. (1.6) or the trace-logarithm of the Green’s function Eq. (1.25) generates a complex analytic function. Having the imaginary part thereby provides complete knowledge of Veff . In this way, the imaginary part (corresponding to field decay by pair creation) and the real part (generating the photon-photon interactions via virtual pair fluctuations) are literally complementary aspects of the field-induced polarization of the vacuum. 26

CHAPTER 2

Key Findings

This chapter brings together important ﬁndings of our study. Extended discussion of methods and results are found in the papers appended.

2.1 Energy-Momentum Tensor of Nonlinear Electromagnetism

In this section, I consider an effective Lagrangian Leff for the electromagnetic fields ~ ~ E, B. Leff must comprise a contribution −S, in accordance with the dominance of Maxwell’s electromagnetism at long wavelength

−1 λ λ¯M = M (2.1) where M is the mass scale rendered from any dimensioned scale displayed by Leﬀ . The energy-momentum tensor T µν is obtained by varying the eﬀective La- grangian with respect to the metric,

µν δLeﬀ T = 2 − gµνLeﬀ . (2.2) δgµν

As for the Euler-Heisenberg Veff (cf. Eq. (1.3) and Sec. 1.4), restricting study to fields satisfying Eq. (2.1) means Leff depends only on the Lorentz scalar and pseudoscalar field invariants 1 1 1 S = F κλF µνg g = (B~ 2 − E~ 2), P = F κλF µνε = −E · B. (2.3) 4 κµ λν 2 4 κλµν written out here to exhibit the presence of gµν in S and its absence in P. Now, variation with respect to gµν produces ∂L ∂L ∂L T µν = − eff (gµνS − F µλF ν ) − gµν L − S eff − P eff . (2.4) ∂S λ eff ∂S ∂P 27

The energy-momentum tensor resulting from the Maxwell Lagrangian −S is manifest as the ﬁrst term in parentheses:

µν µν µλ ν TMax = g S − F F λ (2.5)

µν µ Contracting TMax with gµν shows (Tµ )Max = 0, i.e. Maxwell’s electromagnetism has a traceless energy-momentum tensor. In fact, the format of Eq. (2.4) is designed to separate the traceless component in the first set of parentheses and the energy- momentum trace in the second set, ∂L ∂L T µ = g T µν = −4 L − S eff − P eff . (2.6) µ µν eff ∂S ∂P Now, define (Labun and Rafelski, 2010a) ∂L ε ≡ − eff −→ 1, (2.7) ∂S Leff →−S referred to as the dielectric function and simplifying to unity in the limit of Maxwell’s electromagnetism. In view of Eqs. (2.6) and (2.7), the energy-momentum tensor is distilled to µν µν µν κ T = εTMax + g Tκ /4. (2.8)

κ This format shows that ε and Tκ completely characterize modiﬁcations to the electromagnetic energy-momentum tensor from Maxwell’s electromagnetism. The κ physical signiﬁcance and consequences of Tκ are discussed below in Sec. 2.4. Numeri- cal evaluations of the dielectric function and energy-momentum trace are undertaken in (Labun and Rafelski, 2010a).

2.1.1 4-Momentum and Mass

The energy-momentum tensor describes flow of energy and momentum (Misner et al., 1974). This flow is reduced to a 4-vector by specifying a spacelike hypersurface through which flow is measured, or equivalently an observer with timelike 4-velocity who integrates the flow. The 4-momentum density is

µ µν p = T uν, (2.9) 28

where uν = (1,~0) in the stationary lab frame. Using the lab frame uν, one finds the 0-component E~ 2 + B~ 2 T κ E~ 2 + B~ 2 p0 = ε + κ −→ ; (2.10) 2 4 Leff →−S 2 the Maxwell energy density is modified by ε and the trace makes a new contribution. The spatial components of the 4-momentum density yield i iν ~ ~ p = T uν = ε E × B . (2.11)

The Poynting vector of Maxwell’s electromagnetism,

S~ = E~ × B,~ (2.12) is modified by the dielectric function ε. The magnitude of the total 4-momentum is a Casimir operator of the Poincaré group and is invariant identified as the mass. One first defines the mass density: using the Maxwell limit of Eq. (2.10) along with Eq. (2.12),

r1 √ % = (E~ 2 + B~ 2)2 − |E~ × B~ |2 = S2 + P2. (2.13) Max 4 In the general case, using Eq. (2.8) yields

p µ 2 2 2 κ 21/2 % = p pµ = ε (S + P ) + (Tκ /4) . (2.14)

The mass of the field configuration is the integral of the mass density over the spacelike hypersurface, Z 3 Mfield = d x %. (2.15)

The mass of a plane wave is identically zero, as must be the case given that plane waves are the momentum eigenstates of massless photons. This fact is derived from Eq. (2.14) in virtue of the fact that both invariants vanish S, P = 0 for the plane wave ﬁeld conﬁguration |E~ | = |B~ |, E~ · B~ = 0. (2.16) 29

These facts contribute to the proof that no nonlinear vacuum phenomena occur in the ﬁeld of a plane wave of arbitrary frequency composition (Schwinger, 1951), and consequently, for plane waves the deﬁnition Eq. (2.14) in fact reduces to Eq. (2.13).

2.1.2 Rest Frame of an Electromagnetic Field

The mass Eq. (2.15) selects a preferred frame of reference for the field configuration in which its 3-momentum vanishes. The vector structure of the 3-momentum density Eq. (2.11) implies that in this field rest frame, either a) one of E~ or B~ vanishes, or b) E~ and B~ are parallel or anti-parallel. In case (a), P = 0 and the magnitude of the electric |E~ 0| or magnetic field |B~ 0| is given by

for S < 0, |E~ 0|2 = |S| − S, (2.17) for S > 0, |B~ 0|2 = |S| + S.

For case (b), P = ±|E~ ||B~ |. Continuity requires Eq. (2.17) are the P → 0 limits √ |E~ 0|2 = a2 = S2 + P2 − S, √ (2.18) |B~ 0|2 = b2 = S2 + P2 + S, where a, b are the eigenvalues of the field strength tensor F µν, Eq. (1.10). κ Because Tκ can have an imaginary part originating from Leff , the mass of the field can obtain a finite imaginary part reflecting its ability to decay. As for elementary particles, decay processes are preferentially considered in the rest frame of the decaying electromagnetic field, because in this frame momentum conservation requires the net 3-momentum of all particles resulting from decay to vanish. Indeed, the rest frame of the electromagnetic field must exist in order for decay into massive electrons and positrons to be kinematically allowed. To see this, consider the invariant mass of a single created pair. Evaluating in the center-of-momentum of 30

the frame of the pair, where the electron and positron 3-momentum are equal and opposite ~p− = −~p+ is

2 2 2 2 s = (p+ + p−) = p+ + p− + 2(E+E− − ~p+ · ~p−) = 4m . (2.19)

Nonzero m requires the center-of-momentum frame of the decay products be well- defined and hence also the rest frame of the decaying field. This fact determines the total amount of energy available to convert into pairs and the rest frame of the pairs created by decay of the field. These two consequences are discussed in the next two sections.

2.2 Time Constant for Decay of Electromagnetic Field Mass

The energy available in the field for conversion into pairs is the energy in the field’s rest frame, i.e. the mass of the field configuration Eq. (2.15). The characteristic time for the spontaneous conversion of electromagnetic field energy into electron-positron pairs is the materialization time τ (Labun and Rafelski, 2009). τ is the ratio of the energy available in the field to the rate at which the energy is materialized,

dhE i−1 τ = M mat . (2.20) field dt dhEmati/dt is obtained by weighting the tunneling probability Eq. (1.38) with the energy of the tunneling state before summing over states, see Eq. (4) in (Labun and Rafelski, 2009) for details. The definition Eq. (2.20) produces the standard kinetic theory result for the evolution of field energy

Z t dt0 Mfield(t) = Mfield(0) exp − 0 . (2.21) 0 τ(t ) An electromagnetic field with a prescribed timescale T is quasi-constant for the purposes of treating dynamics of spontaneous pair creation when

T τ. (2.22) 31

In the quasi-constant limit T → ∞, the only scale present is the electron mass, and it is natural to expect that τ ∼ m−1. In fact, the leading behavior is

~ ~ α m τ ' τ eπ|Ec|/|E|, τ −1 = = 1.4 1017 s−1, (2.23) mat mat π 4π with corrections becoming noticeable only for fields near to the critical magni- ~ tude Eq. (1.1). The exponential suppression by |Ec| in Eq. (2.23) arises because spontaneous emission of pairs is a semi-classical tunneling process, as explained in Sec. 1.5.3. Spontaneous decay is consequently a slow process for field strengths field ~ below the critical magnitude |Ec|. The materialization time τ is faster than the vacuum decay rate per unit vol- ~ ume 2Im Veff Eq. (1.31) for fields up to ∼ 0.5|Ec| where corrections to the leading exponential behavior begin to set in (Labun and Rafelski, 2010c). When ultra-strong fields are created in the lab using high intensity lasers, the typical timescale of the field is at or below the femtosecond (10−15 s) scale. A ~ −12 laser field attaining 0.2|Ec| has τ = 2.8 10 s; materialization is too slow to reduce ~ noticeably the total energy in the laser fields. Near-critical fields ∼ |Ec| are necessary before materialization is a visible cause of energy loss in attosecond-length pulses. For pulses of many pico- or femtosecond length, our results show the consistency of the assumption of slow variation Eq. (2.22), because for a field approaching the critical magnitude, τ is 1000 times shorter than the laser timescale.

2.3 Energy of Particles from Laser-Induced Vacuum Decay

The rest frame of pairs created from decay of the field must coincide with the rest frame of the decaying field identified in Sec. 2.1.2. In view of the manifest dependence of the field 3-momentum Eq. (2.11) on the electric and magnetic field geometry, one can discuss manipulating the motion of the field’s rest frame. In particular, by choosing the field’s rest frame to have high velocity relative to the lab, high energy electron-positron bunches can be created without accelerating charged 32

particles. For long wavelength laser ﬁelds, one may consider the mass density and use the decay per unit volume per unit time. Being a function of time and space, the mass density Eq. (2.15) determines an instantaneous and local rapidity yS of the ﬁeld’s moving rest frame

(p0)2 S~2 %2 + ε2(E~ × B~ )2 cosh2 y ≡ = 1 + = . (2.24) S %2 %2 %2

The last expression exhibits the dependence of yS on the geometric relation between the electric and magnetic field vectors through the Poynting vector S~. As for particles, large electromagnetic momentum density |S~| > % in a given frame implies high rapidity. A high-intensity laser pulse has large Poynting vector S~ as measured in the lab and hence large field momentum. Moreover, a weakly-focused pulse contains has a small invariant mass Eq. (2.15), because its fields are near to but not exactly the plane wave configuration Eq. (2.16) and its field invariants S, P are small. In view of Eq. (2.24), large Poynting vector and small invariant mass mean that a weakly- focused laser pulse has a rest frame which is at high rapidity relative to the lab frame. If electron-positron pairs were to be produced via spontaneous decay in such a laser pulse, the pairs would be materializing in a frame traveling at high velocity relative to the lab. However, the same fact that the fields are nearly of the planewave form Eq. (2.16) also leads to a small field invariant a Eq. (1.10) and suppresses the rate of vacuum decay, recall Eq. (1.31). A high rapidity of the field rest frame can still be taken advantage of by colliding two laser pulses. The frame in which the momenta of the pulses balance is the center- of-momentum frame of the collision, and this frame coincides with the rest frame of the total electromagnetic field

~ ~ ~ ~ ~ ~ Etot = E1 + E2, Btot = B1 + B2, (2.25) 33

obtained as the superposition of the individual laser fields. The motion of the center- of-momentum frame relative to the lab is controlled by the field strengths of the laser pulses and their collision geometry, as demonstrated by example now. Consider two linearly polarized laser pulses converging at an angle θ (Labun and Rafelski, 2011). The wavelength of optical lasers is 1240 λ ' 1054 − 264 nm, i.e. photon energies ω = = 1.18 − 4.71 eV, ~ λ and hence are quasi-constant, cf. Eq. (1.3). The fields in the pulse are modeled as constant fields satisfying Eq. (2.16), localized by step-function cutoffs in both transverse and longitudinal directions. In the case of aligned laser polarizations, the total electric field is independent of θ, while the direction of the total Poynting ~ ~ ~ vector Stot = Etot ×Btot depends on θ through the vector sum of the magnetic fields. The motion of the center-of-momentum frame is rotated from the beamline of the stronger laser pulse by an angle δ given by r sin θ |E~ | tan δ = , r = 1 ≤ 1 (2.26) ~ 1 + r cos θ |E2| Here, the weaker laser field is denoted with the 1 subscript. When r → 1, the momenta of the laser pulses are equal and accordingly δ → θ/2. The mass density also depends on the collision geometry: by choosing the two ~ ~ laser pulses to have aligned, linear polarizations, Ptot = −Etot · Btot = 0, and ~ 2 ~ 2 Btot − Etot is controlled by the convergence angle θ, which is also the angle between ~ ~ B1 and B2. Only the electric-like invariant a Eq. (1.10) is non-zero,

2 ~ 2 2 a = 2r(1 − cos θ)|E2| , b = 0. (2.27)

The squared mass density,

2 ~ 4 2 2 2 κ 2 % = 2|E2| ε r (1 − cos θ) + (Tκ /4) , (2.28) decreases with decreasing θ. %2 vanishes in the collinear limit θ → 0, recovering a superposition of massless plane waves. The rapidity of the moving rest frame of the 34

ﬁeld is ! !−1 r−1 + r + 2 cos2(θ/2)2 T κ 2 sinh2 y = − 1 1 + κ . (2.29) S 2 sin2(θ/2) 4εa2

The rapidity increases with decreasing θ, diverging in the θ → 0 limit. Together with Eq. (2.26) and Eq. (2.28), this equation shows how the rapidity of the field rest frame and mass density are controlled by the geometry of the collision. As discussed in Sec. 2.1.2, the rest frame of electron-positron pairs created by vacuum decay coincides with the field rest frame. The energy of the pair in the field rest frame is suppressed by the Gaussian tunneling probability Eq. (1.39). The high rapidity of the field rest frame relative to the lab boosts the decay products in the ~ direction of Stot. The pairs are therefore observed in the lab frame with a rapidity

distribution peaked around the high rapidity yS. Qualitative characteristics of the spectrum of particles produced in the modeled collision are discussed in (Labun and Rafelski, 2011). The spectrum is determined by the particles’ origin in vacuum decay. Because the laser fields satisfy the long- wavelength condition, modeling the electromagnetic fields arising in the collision in greater detail is expected to have a small effect on the spectrum. The large ~ boost in the direction Stot helps to distinguish products of vacuum-decay in laser pulse collisions from particle-creation backgrounds such as cascades and perturbative processes. This salient characteristic of the spectrum thereby enhances discovery potential of the vacuum decay phenomenon by selecting the geometry of the laser pulse collision.

2.4 The Energy-momentum Trace

An energy-momentum trace signals the presence of a dimensioned scale. Maxwell’s µν electromagnetism is a scale-invariant theory: its Lagrangian LMax = −F Fµν/4, Eq. (1.11), is a combination of the electromagnetic ﬁelds in which no dimensioned κ parameter is required, and the Maxwell Tκ is zero, cf. Eq. (2.6). Integrating the 35

quantum fluctuations of the electron introduces m as the intrinsic scale of the fluctuations. The explicit presence of the electron mass implies the energy-momentum tensor of the effective electromagnetic theory has a nonzero trace. The relationship of the scale m to the energy-momentum trace are made visible by writing (1) 4 S P Leff = −S + V + m Vfeff , . (2.30) eff m4 m4 (1) Here, the Maxwell Lagrangian is separated from Veff , which comprises terms linear in S (quadratic in E,~ B~ ), such as the charge normalization arising in QED, Eq. (1.26).

The dimensionless Vfeff contains new contributions nonlinear in S, P, for example the finite, renormalized Euler-Heisenberg effective potential rescaling the u-integral in 4 Eq. (1.9) so as to write Vfeff = Veff /m . κ Recalling the format of Tκ Eq. (2.6), one sees that ! κ d 4 ∂Vfeff ∂Vfeff T = −m m Vfeff = −4 Vfeff − S − P . (2.31) κ dm ∂S ∂P

κ Terms linear in S make no contribution to Tκ being removed by Vfeff −S(∂Vfeff /∂S). In (1) κ particular, Veff does not contribute to Tκ , even though it may depend on the scale m. (1) In case Veff does depend on m, the correct mass dimensionality is preserved by the presence of at least one other scale additional to m. The generalization of Eq. (2.31) (1) κ to two scales confirms the absence of a contribution by Veff to Tκ . The energy- momentum trace thus arises from the intrinsically nonlinear interactions allowed into the electromagnetic theory, i.e. diagrams corresponding to photon-photon scattering seen in figure 1.2. This result is already seen in the work of Adler et al.(1977). For effective potentials obtained as quantum corrections such as the Euler-

Heisenberg Veff , the relation Eq. (2.31) has important consequences for the numerical κ ¯ value of Tκ , specifically relative to the Fermi condensate hψψi. For this reason, I review in the next subsection the definition of the condensate hψψ¯ i and its rela- κ κ tion to Tκ . For the Euler-Heisenberg Veff , Tκ is greater than or equal to zero in 36

κ most electromagnetic field configurations. Tκ only becomes negative in very strong ~ ~ electric fields, |E| & 9|Ec|, for which the field also has a short lifetime due to spontaneous pair creation, cf. Sec. 2.2. As discussed below (Sec. 2.4.2), this positivity of κ Tκ supports the role of the energy-momentum trace as an excitation energy of the vacuum in the presence of external fields.

κ 2.4.1 Fermi Condensate in Vacuum and Tκ

A measure of the strength of the quantum fluctuations is the electron-positron condensate hψψ¯ i. The condensate is derived by comparing normal ordering the field operators in the with-field vacuum to the no-field vacuum. To see this, consider the Wick decomposition of time-ordered products,

Tb ψ(x0)ψ¯(x) =: ψ(x)ψ¯(x0) : + h0 |Tb ψ(x)ψ¯(x0) | 0i. (2.32)

Taking the expectation of both sides in the with-ﬁeld vacuum h...i,

h: ψ(x)ψ¯(x0):i = hTb ψ(x)ψ¯(x0)i − h0 |Tb ψ(x)ψ¯(x0) | 0i (2.33) 0 0 = iG(x, x ; Aext) − iG(x, x ; 0), shows that normal-ordering in the with-ﬁeld vacuum is equivalent to computing the change in the propagator resulting from the presence of the ﬁeld. In the x0 → x limit, the left hand side is the condensate, which is conventionally written with the normal-ordering symbols : : omitted and the Dirac trace implied upon commuting the operators, h: ψ(x)ψ¯(x):i → −hψψ¯ i(x).

The right side of Eq. (2.33) is related to Veff in virtue of dL m eff = im tr [G(x, x; A ) − G(x, x; 0)] (2.34) dm ext which follows from Eq. (1.25). The same ξ-limit as Eq. (1.21) is implied for equal spacetime arguments in the propagator. Together Eqs. (2.33) and (2.34) imply dL m eff = −mhψψ¯ i. (2.35) dm 37

Comparing this identity to dLeff /dm obtained from Eq. (2.30), one sees that dV (1) T κ = m eff + mhψψ¯ i, (2.36) κ dm a relation found by Adler et al.(1977). The energy-momentum trace is thus the condensate after compensating its leading term in S. The condensate is negative in dominantly electric fields and positive in domi- (1) nantly magnetic fields due to the leading behavior from S. Adding dVeff /dm in κ Eq. (2.36) ensures that Tκ is positive for most electrical fields. Non-perturbative numerical evaluations of the condensate and energy-momentum trace generated by the Euler-Heisenberg Veff are found in (Labun and Rafelski, 2010a,b).

κ 2.4.2 Astrophysical Eﬀects of Tκ

An energy-momentum trace has gravitational effect opposite to that of normal mat- µ ter. Consider the perfect fluid Ansatz for the energy-momentum tensor Tν = diag(ρ, −p, −p, −p). For normal matter, 3p ≤ ρ, with equality obtained for rel- µ µ κ ativistic particles or radiation. In comparison to normal matter, Tν = δν Tκ = κ κ κ κ diag(Tκ ,Tκ ,Tκ ,Tκ ) appears as energy density with pressure of the wrong sign. The reversal of gravitational influence is visible in the trace of the Einstein equation. For example, in a cosmological setting such as the Friedmann metric (Kolb and Turner, 1994) a¨ 3 = −4πG(ρ + 3p) + 2πGT κ + Λ. (2.37) a κ a positive energy-momentum trace drives acceleration in the same manner as a cosmological constant. In contrast, gravitating dust makes a negative contribution, causing deceleration. Out to redshift z = 1, the acceleration of the universe is found to be consistent with a source that is independent of time, such as a cosmological constant or vacuum energy (Serra et al., 2009). This acceleration corresponds to a dark energy density, Λ d ' (2.325meV)4 ' 6.09 10−10J/m3. (2.38) 4πG 38

Explaining the observed Λd as an energy-momentum trace arising from the Euler-

Heisenberg Veff requires fields very strong by laboratory standards: an electric or magnetic field of respectively

~ 9 −8 ~ ~ −8 ~ |E|d = 32.4 10 V/m = 2.5 10 |Ec|, |B|d = 108 T = 2.5 10 |Bc| (2.39) produce an energy-momentum trace equal to the dark energy density Eq. (2.38). The measured intergalactic magnetic field on the order of 10−10 T(Giovannini, 2004) is insufficient to explain the observed dark energy. µ To investigate the effect of localized domains with Tµ , one analyzes the Oppenheimer-Volkoff equations determining the structure of a self-gravitating object. Inspection of the differential equations as well as numerical integrations show κ that presence of Tκ along with normal matter leads to more massive objects. How- κ ever, the magnitude of the QED Tκ is too small to noticeably influence the structure of the expected post-main sequence star. Even in the most extreme possibilities that 12 3 ~ κ 3 4 stellar magnetic fields achieve 10 T ∼ 10 |Bc|, Tκ reaches only ∼ 10 m . With its scale set by the electron mass m4 = 8.9 10−9 MeV/fm3, the QED energy-momentum trace remains too small in comparison to pressures predicted in nuclear matter p ∼ 1 MeV/fm3. Outside the star, the pressure of matter is much smaller and a κ force arising from Tκ is compared directly to gravity, which is studied in the next section.

2.4.3 Modiﬁed Lorentz Force

Seeing that an energy-momentum trace has gravitational effect opposite to that κ of normal matter and that Tκ arises from the effective nonlinear field-field inter- κ actions, the question arises how Tκ modifies the dynamics of charged particles in strong external fields. In (Labun and Rafelski, 2010b), the Euler-Heisenberg effective potential is shown to generate a correction to the Lorentz force in the classical field theory of electromagnetism and the result is applied to a charged particle in the environment of a strong external (stellar) magnetic field. 39

The interaction energy-momentum is identified covariantly as the component of µν the total electromagnetic energy-momentum tensor Tt left over after removing the µν µν contributions depending only on the external field Te and the particle field Tp :

µν µν µν µν Tt = Te + Tp + Tint . (2.40)

~ µν Here, the particle’s ﬁeld is small in comparison to |Ec| and Tp can be taken as the Maxwell energy-momentum tensor. The particle has also inertial energy-momentum µν Tinertial derived from its mass. Enforcing conservation of total energy-momentum then deﬁnes the total 4-force f µ due to electromagnetic interactions as compensating the change in inertial momentum of the particle. The 4-force is the divergence of the interaction term,

µν µ µν µ − ∂νTint ≡ f = jνFe + δf . (2.41)

µν p µν The standard Lorentz force −∂νTep = jν Fe is found as the leading contribution. The Euler-Heisenberg-induced correction is

κ µ 2 µν p µν 2 µ 2 − ε ∂Tκ e αβ δf u (εe − 1) Fe jν − Tep ∂ν(εe − 1) − ∂ FαβKp . (2.42) 4 ∂S e

µν Equality is only approximate, due to two expansions: first, Tt has been expanded ~ ~ to first order in the particle field with respect to the strong external field, |Dp|/|Be|, and the e subscripts indicate the coefficients are to be evaluated at the external field. Second, the relation Kµν(F µν) Eq. (1.13) has been expanded to second order in α. An advantage of this formulation is the dependence of the result on only the Maxwell energy-momentum tensor, the dielectric function ε and energy-momentum κ trace Tκ . The Euler-Heisenberg-generated correction has significant consequences for the particle dynamics near a strongly-magnetized compact star because it shows energy can be transferred to the particle by a strong magnetic field. The last term of Eq. (2.42) is responsible for the new repulsive component of the force, and within 40

µ this term Tµ makes the largest contribution. The magnitude of the usual magnetic ~ component ~v × Be of the Lorentz force is of course much larger than the correction Eq. (2.42), but is always transverse to the motion of the particle and can do no work to expel the particle from the field. µ With the help of the non-perturbative evaluations of ε and Tµ , I quantitatively µ compare the strength of the Tµ -induced repulsive force to the gravity of a compact star, see Figure 2 in (Labun and Rafelski, 2010b). The force Eq. (2.42) significantly affects matter accretion and stellar collapse dynamics in astrophysical environments −4 5 where strong magnetic fields in excess of B 10 Bc = 10 T are known to exist (Harding and Lai, 2006).

2.5 Comparison of Modiﬁcations of the Maxwell T µν

It is informative to compare the dielectric function ε and energy-momentum trace κ Tκ resulting from the Euler-Heisenberg eﬀective potential Eq. (1.9) to their values in 1) the limiting ﬁeld-strength extension of Maxwell’s electromagnetism proposed by Born and Infeld(1934), and

2) the Euler-Heisenberg Veff obtained for a scalar (spinless) electron. κ In (Labun and Rafelski, 2010a), ε and Tκ are evaluated numerically up to the limiting field strength in Born-Infeld electromagnetism and for field strengths covering 3 orders of magnitude above and below the critical magnitude Eq. (1.1) for the

Euler-Heisenberg Veff of a Dirac and scalar electron. As a resolution to the apparently infinite energy in the electromagnetic field of the point-like electron, Born and Infeld(1934) introduced a modified theory of ~ electromagnetism with an explicit upper limit |EBI| on the attainable field intensity. Implementing this limiting field strength via a mass scale, in analogy to Eq. (1.1),

~ 2 |EBI| = M /e, (2.43) 41

the Born-Infeld Lagrangian is 4 p 4 4 2 LBI = M 1 − 1 + 2S/M − (P/M ) . (2.44)

κ The integrand is a smooth analytic function of S, P. Consequently, ε and Tκ are also smooth functions of S, P: Differentiation of Eq. (2.44) with respect to S yields ∂L ε ≡ − BI = 1 + 2S/M 4 − (P/M 4)2−1/2 , (2.45) BI ∂S which diverges approaching the limiting field strength. The Born-Infeld Lagrangian contains no terms linear in S, and the energy-momentum trace is obtained by directly differentiating with respect to M, s ! ∂L 1 + 2S/M 4 + (S/M 4)2 (T κ) = −M BI = 4M 4 − 1 ≥ 0. (2.46) κ BI ∂M 1 + 2S/M 4 − (P/M 4)2

The fact that the energy-momentum trace is positive also in the Born-Infeld theory lends added support to its consideration as the excitation energy of the with-field vacuum. κ The smoothness of the functions εBI and (Tκ )BI in the SP-plane contrasts with κ the box-like structure of ε and Tκ found in the quantum theories: In Figure 1 κ of (Labun and Rafelski, 2010a), εBI and (Tκ )BI display a characteristic square root structure derived from the limiting field strength and the zero of the root 2 in Eq. (2.44), 1 + 2S = P . For the Euler-Heisenberg Veff of the Dirac and scalar κ electron, Tκ and ε changes suddenly from mostly constant in S to mostly constant in P. The lines in the SP-plane along which the change occurs differs from the Dirac to the scalar electron. In (Labun and Rafelski, 2010a), compare Figures 3 and κ 5 for Tκ , and Figures 6 and 7 for ε. In dominantly magnetic fields (S > 0), the dielectric shift ε − 1 is negative for all three Lagrangians, the Born-Infeld LBI Eq. (2.44) and the two Euler-Heisenberg effective potentials for the Dirac and scalar electrons. ε thus suppresses the Maxwell tensor in Eq. (2.8). For the Euler-Heisenberg potentials, the effect is small with 42

−3 ~ |ε − 1| < 10 up to ﬁeld strengths exceeding critical |Bc|. For Born-Infeld theory however, this relative suppression can signiﬁcantly enhance observable consequences of the presence of the energy-momentum trace.

2.6 Dark Energy due to a Metastable Vacuum State

The universe may reside today in a metastable, excited vacuum state. In this scenario, the cosmological dark energy is identified as the excitation energy (Labun and Rafelski, 2010d). Sec. 2.4.2 showed how the role of a vacuum (excitation) energy κ can be played by a positive energy-momentum trace Tκ . Dark energy being a long wavelength phenomenon, the Euler-Heisenberg field-induced vacuum structure offers κ a model generating Tκ from long-wavelength external fields, in which to investigate quantitatively the energy scales of interest. 4 For m large compared to Λd/4πG Eq. (2.38), a field weak on the scale set by κ m is both stable and providing a sufficient Tκ . In the Euler-Heisenberg Veff , an ~ −8 ~ electric field |E|d ∼ 10 |Ec|, cf. Eq. (2.39), generates an energy-momentum trace comparable to the dark energy density. On the other hand, using Eq. (2.23), the associated timescale for the decay of such a field is τ = 1054 years, safely exceeding ~ the present age of the universe. The fact that |E|d is large by laboratory standards is attributable to the mass scale of QED being m = 511 keV. Considering the mass scale m a free parameter, I now ask what m admits a field having a lifetime greater than the age of the universe while generating the observed √ dark energy. For e = 4πα fixed to the QED coupling, the strength of the field must remain below m2/e|E~ | ' .04 for the excited state to persist over the history of the universe. Up to factors of order unity, m should be in the range 1 meV – 1 eV. This suggests that dark energy could be related to the unresolved vacuum degeneracy in quantum chromodynamics associated with CP symmetry or neutrinos and their masses (Bjaelde and Hannestad, 2010). The hypothesis that an excited vacuum state generates the dark energy can be 43

veriﬁed by observation of a release of the excitation energy associated with transition to the true vacuum. In this context, studying in general terms vacuum structure and decay in strong ﬁelds may be providing insight into latent and hidden structure giving rise to the cosmological dark energy.

2.7 Summary and Conclusions

This dissertation has introduced work in several research publications describing certain advances in understanding of strong ﬁeld QED physics. Ultra high intensity laser pulses and vacuum structure in astrophysics, two domains of intense current interest, have guided the investigation, and here the key results of importance to these ﬁelds are summarized in brief:

Laser pulses:

1. Section 2.1.2, see (Labun and Rafelski, 2009) The energy available for conversion into electron-positron pairs is the field energy in the frame of reference in which the Poynting vector vanishes. This frame is also the center-of-momentum frame of the created electron-positron pairs. I derive in covariant form the energy available for materialization of pairs, i.e. the mass of a field configuration, for arbitrary (effective) nonlinear theory of electromagnetism Eq. (2.15).

2. Section 2.3, see (Labun and Rafelski, 2011) The motion of the mass of the electromagnetic field configuration in the laboratory determines the motion and energy of the electron-positron pairs created by decay of electromagnetic fields. This simple insight allows characterization of the produced pair spectra. I present results for the case of two nearly collinear converging laser pulses. 44

Astrophysics:

3. Section 2.4.3, see (Labun and Rafelski, 2010b) For a charged particle in a long-wavelength external field, the effective QED nonlinear field-field interactions induce a new radially-directed force in a magnetic (dipole) field Eq. (2.42). This force ejects a moving charged particle from the external field and is strong enough to compete with the gravity of a compact stellar object. This additional effective force includes contributions from the (vacuum) polarization dielectric function Eq. (2.7) and energy-momentum trace Eq. (2.6).

4. Section 2.6, see (Labun and Rafelski, 2010d) The cosmological dark energy could be the energy of an excited vacuum state arising from yet-to-be-determined non-perturbative structure in the vacuum. Such an excited state must have a lifetime greater than the age of the universe considering both static and induced processes leading to decay of the excited state. I use the Euler-Heisenberg type ﬁeld-induced vacuum structure in QED as a model to explore consistency of these requirements, resulting in a mass- energy scale 1 meV − 1 eV.

The renaissance of strong field QED is driven by theoretical techniques greatly improved in the context of strongly interacting gauge theories, in particular QCD. This has lead to improved understanding of the non-perturbative quantum vacuum structure. In a fortuitous coincidence, laboratory experimental opportunities have recently arisen involving high intensity lasers. Future efforts in phenomeno- logical strong field QED are likely to focus on the new physics phenomena arising in laser pulse interactions and materialization of laser energy into particle pairs.

The Euler-Heisenberg Veﬀ Eq. (1.9) provides an analytical stepping stone regarding both particle production and the understanding of the vacuum structure of quantum theories, leading on to description of dark energy as a vacuum phenomenon. 45

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APPENDIX A

Energy of the Dirac Vacuum

Eq. (1.5) is obtained by considering the explicitly charge symmetric Hamilto- nian (Rafelski et al., 1978) Z 1 Hˆ = d3x ψ,¯ (iγkD + m)ψ + ψ¯(iγkD∗ + m), ψ . (A.1) 4 k k in which the Dirac operator in the second commutator acts to the left. The sum over suppressed spinor indices is implied by writing ψ¯ on the left. As shown below, symmetrizing the Hamiltonian in this way ensures the zero-particle state is an exact charge eigenstate with total charge zero. One expands the ﬁeld operator in particle (positive frequency n > 0) and anti- particle (negative frequency n < 0) contributions

X ˆ X ˆ† ψ = bnψn + dnψn. (A.2) n>0 n<0 Inserting this expansion into Eq. (A.1), 1 X h i 1 X h i Hˆ = ˆb† , ˆb − dˆ† , dˆ , (A.3) 2 n n n 2 n n n n>0 n<0 shows the Hamiltonian is manifestly positive,

ˆ X ˆ† ˆ X ˆ† ˆ H = nbnbn − ndndn + Usea, (A.4) n>0 n<0 since the negative sign in front of the second sum combines with the sign of the energy eigenvalue n < 0 to give a positive contribution. Now taking the expectation value in the zero particle state, ! 1 X X U = h0 |Hˆ | 0i = − − (A.5) sea 2 n n n>0 n<0 exhibits symmetric contributions from the particle and anti-particle states. In the absence of external electromagnetic potential, Usea in Eq. (A.5) corresponds to Usea(0) in the introduction. 49

The total charge operator is the zero-component of the current operator integrated over space e Z Qˆ = d3x ψ†, ψ . (A.6) 2 Upon inserting the expansion Eq. (A.2), ! ! X X e X X Qˆ = e ˆb† ˆb − dˆ† dˆ + − (A.7) n n n n 2 n>0 n<0 n<0 n>0 with the sign diﬀerence between particle and anti-particle sums arising from their respective charges. The total charge vanishes in the zero particle state h0 |Qˆ | 0i = 0, provided equal numbers of states constitute the sums over positive and negative eigenvalues in the second pair of parentheses. Moreover, Qˆ commutes with the charge symmetrized Eq. (A.3), so that there exist simultaneous eigenstates of the Hamiltonian and total charge. 50

APPENDIX B

Vacuum Decay Time in Strong External Fields

Vacuum Decay Time in Strong External Fields Labun, L. and Rafelski, J. Physical Review D79 (2009) 057901 doi: 10.1103/PhysRevD.79.057901 arXiv:0808.0874 [hep-ph]

Copyright (2009) by the American Physical Society Reprint permission granted under APS Copyright Policies: http://publish.aps.org/copyrightFAQ.html#thesis 51

PHYSICAL REVIEW D 79, 057901 (2009) Vacuum-decay time in strong external ﬁelds

Lance Labun and Johann Rafelski Department of Physics, University of Arizona, Tucson, Arizona, 85721, USA and Department fu¨r Physik der Ludwig-Maximillians-Universita¨t Mu¨nchen und Maier-Leibniz-Laboratory, Am Coulombwall 1, 85748 Garching, Germany (Received 7 August 2008; published 2 March 2009) We consider dynamics of vacuum decay and particle production in the context of short pulse laser experiments. We identify and evaluate the invariant ‘‘materialization time,’’�, the time scale for the conversion of an electromagnetic ﬁeld energy into particles and compare to the laser related time scales.

DOI: 10.1103/PhysRevD.79.057901 PACS numbers: 12.20.Ds, 11.15.Tk, 42.50.Xa

In the past decade high intensity short pulse laser tech- strength as are present in the laser pulse, a phenomenon nology has advanced rapidly [1], pulses achieved inten- used in laser-ion acceleration [33]. We thus address in this sities of 1026 W=m2 [2,3]. With subsequent concentration work the general circumstance of a spatially homogeneous by coherent harmonic focusing allowing a further gain electrical field. in intensity of around 6 orders of magnitude [4], In all laboratory experiments supercritical fields (fields laser technology is nearing the scale of rapid vacuum capable of spontaneous particle production) will be 2 33 2 instability, c�0E0=4� 4:65 10 W=m , whereE 0 strongly time dependent. One must distinguish two pro- m2c3=e@ 1:32 10 ¼18 V=m�. The study of vacuum insta-� foundly different experimental regimes involving vacuum bility with¼ laser� pulses involves dynamics on a time scale rearrangement: set by the pulse length, which at optical frequencies im- (a) the field is established on a time scale much faster 15 plies that the fields are in existence for 10 � s and may than the typical vacuum-decay time, such as was 18 � reach 10 � s when coherent harmonic focusing is used. studied in heavy ion collisions [8,34], with the spec- The� vacuum state of quantum electrodynamics (QED) is trum of produced particles (positrons) representative metastable in the presence of electrical fields of any of the single particle states achieved with the ulti- strength, but only in proximity ofE 0 does the effect occur mate field strength; on an observable time scale [5,6], as we exhibit below. (b) the field is established on a time scale slower than Specifically, we investigate whether the laser pulse time the decay time of the vacuum (adiabatic switching). scale allows the vacuum in strong fields to relax, thereby In this case a particle will be produced just upon admitting the new vacuum to experimental investigation achievement of supercriticality and thus at zero using pulsed lasers. The dynamics of ‘‘false’’ vacuum (longitudinal) momentum. This is the assumption decay have been studied in the context of spontaneous under which the EHS instability is derived. positron creation in heavy ion collisions [7–9] and cosmological models [10–12]. The QED vacuum decay has not In both cases the Fourier-frequency spectrum of the field been directly observed in heavy ion collision experiments, formation assists the process of vacuum emission of parti- due to the relatively long time scale of vacuum-decay cles, in which case we speak of induced (as compared to dynamics as compared to competing processes. However, spontaneous) vacuum decay [35–38]. particle production in strong fields has found a fertile field We evaluate the rate per unit of time and volume of field in quantum chromodynamics [13,14]. energy materialization in the adiabatic EHS switch-on Considerable effort went into generalizing the Euler- limit by calculating the tunneling probability in the pres- Heisenberg-Schwinger (EHS) [5,6] pair production ence of the local potential generated by a (nearly) constant - mechanism for a variety of large-scale (compared to� e field [13]. One starts with the action of an electron with 13 ¼ 2 2 2 @=mec 3:86 10 � m) space- and time-dependent transverse energy� p m in the inverted (i.e. field configurations¼ � [15–17] and to incorporating backre- Euclidean time) potential? ¼ ? þ action [13,18–21]. A stable, modified vacuum state has � =eE � =eE ��2 ? ? 2 2 only been obtained when the field fills a finite space-time S p~ dz dz � eEz ? ; (1) ¼ 0 j j ¼ 0 � ð Þ ¼ 4eE domain [18]. The perturbative vacuum is also stable for an Z Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi? ideal plane wave (laser) field of arbitrary strength, and thus with the upper bound determined by the turning point in many investigations have focused on understanding pair the potential, at which the quasi-longitudinal momentum production in optimized pulsed laser field configurations (and therefore the pair) becomes real. The tunneling proba- [22–32]. More recently it has been also noted that in the bility for the electron-positron pair is then twice the action interaction of laser pulses with thin foils, the charge sepa- in Eq. (1) ration effect due to a much greater electron mobility helps 2 2 �=eE �2 in achieving longitudinal electrical fields of comparable � � exp 2S e �ð Þ ? : (2) ð ?Þ ¼ j ð� Þj ¼

1550-7998=2009=79(5)=057901(4) 057901-1� 2009 The American Physical Society 52

BRIEF REPORTS PHYSICAL REVIEW D 79, 057901 (2009) 2 By integrating s � 1 � � over alls,p one This is the central result of this paper in its simplest ? ½ � ð ?Þ� ? qualitative form, shown in Fig. 1 as the upper (red) line. reproduces the SchwingerQ Q series expression for the total vacuum persistence probability, as was ﬁrst noted by Materialization time for E

d um eE 1 2 �=eE �2 purely electric field. h i 2 d� 2� e�ð Þ ? : (5) dt ¼ 2� m ? ? We next evaluate� adding a constant, homogeneous Integration by parts resultsZ in magnetic field. The energy available in the presence of both electric and magnetic fields is evaluated in the local d u �E0 m ! E2e �E0=E 1 h ; (6) rest frame from the four-vector of energy momentum h i 0 ð� Þ �� dt ¼ � þ �sffiffiffiffiffiffiffiffiffiE �� T v� (see e.g. [44]) i.e.u f T v� . This ‘‘mass 2 - 2 2 ¼ j j in which! 0 : �c=� �e �mc =� @ 5:740 density’’ of the field is expressed in terms of the in- ¼ 2 ¼ ¼ � 1017 s 1 andh z : p�ez erfc z =2z: The asymptotic variantsF 1=4 F ��F 1 B2 E 2 andG � ð Þ ¼ ð Þ ¼ð Þ �� ¼ 2 ð � Þ ¼ behavior of the complementary error function implies 1=4 F��F E B as ffiffiffiffi ð Þ �� ¼ � thath z increases linearly with E=E 0, specifically, forz ð Þ � 2 2 2 2 1,h z E=4E . uf F G f F;G A F;G ; (9) ð Þ� 0 ¼ ð þ Þ ð Þþ ð Þ The relaxation time of the metastable with-field vacuum wheref @L=@qFffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 (Maxwell) andA is the confor- ¼ !� � state via materialization is the ratio of the available supply mal anomaly induced by external fields,T � 4A[ 45]. of field energyu f (density) to the rate of electromagnetic ¼ field energy conversion Eq. (6): 1e-06 d u 1 : m � τ � u f h i : (7) (0) ¼ dt 1e-09 τ � � −1 ω /4 We assume that the pairs decohere rapidly so that the 0 reverse process is impossible, and Eq. (7) then generates 1e-12 the usual kinetic result τ [ s ] t dt =� t 0 1e-15 u t u t 0 e� 0 ð Þ: ð Þ¼ ð Þ R � provides the time at which all field energy is converted into mass, and hence we refer to it as the materialization 1e-18 time of the field. A rough time scale may be obtained by ignoring the 1.0 1 01 E / E second term Eq. (6) and the nonlinear corrections in the 0 2 field energy, usingu f E =2: ¼ FIG. 1 (color online). Materialization time� (solid line) and 1 !0� 0 0 : �E0=E �ð Þ (lighter, red line) as a function of the external normalized �ð Þ eð Þ: (8) 1 2 2 ¼ 4 field E=E . Dashed line:! � =4 � @=4�mc 0:435 as. 0 0 ¼ ¼

057901-2 53

BRIEF REPORTS PHYSICAL REVIEW D 79, 057901 (2009) We remark at this point thatd u =dt is Lorentz invari- 1e-06 h mi ant. It follows that� as defined is the Lorentz invariant -2 1e-08 B/E = 10 (proper) decay time of the vacuum, and we may choose a B/E = 1 suitable frame in which to evaluate�. WhenE B 0, 1e-10 2 pairs are produced whenever the generalized electric� � field B/E = 10 a F 2 G 2 F is nonzero, and the rate reduces to 1e-12 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ � � τ [ s ] that ofp Eq.ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (8). ForE B 0, a reference frame exists 1e-14 whereE andB are either� parallel or antiparallel. Carrying through the tunneling calculation in this frame, we observe 1e-16 the quantization of the transverse momentum of the final states: 1e-18

�=eE �2 1.0 1 01 �l;r e �ð Þ l;r ; E / E ¼ 0 2 with� l;r the energy eigenvalues in the combined field, FIG. 2 (color online).� for several ratios of B=E as a function m2 2eBl r 1; �2 þ ¼� (10) of E=E0, evaluated according to Eq. (13), using the full nonlinear l;r ¼ m2 2eB l 1 r 1; u of Eq. (9). � þ ð þ Þ ¼þ f in whichl 0;1;2;... The integration over transverse nature which should be well observable. If the field energy momenta converts¼ into the usual sum over Landau levels, is in kJ range, this implies that 0.15 J materialize, which d um eE eB �=eE �2 greatly exceeds the energy converted into particles ð Þð Þ � e�ð Þ l;r : (11) h i 2 l;r O 10 erg in most extreme laboratory particle collision dt ¼ 2� l;r ð Þ X reactions.ð Þ Thus, for experiments alert to pair production, Noting the double degeneracy of all but thel 0 mode, we actually attaining the critical fieldE is not necessary for a ¼ 0 sum overr and apply the Euler-Maclaurin summation clear signal of vacuum-decay (materialization) processes. formula. This produces the convergent form In the context of relativistic focusing, where the characteristic time scale is expected to be as short as a few atto- d u �E 18 m 2 �E0=E 0 2 seconds (10� s), the critical field strength needs to be h i ! 0E e�ð Þ 1 h � B ;E ; dt ¼ � þ �sffiffiffiffiffiffiffiffiffiE � � ð Þ� reached to achieve spontaneous vacuum decay (as opposed 2k 2k 1 to induced pair creation). E0 1 B2k 2B d � � : e �E0=E We further note that in the above example the percentage E 2k ! E ð Þ 2k 1 ¼ k 1 � 0 � dx � X¼ ð Þ of the field energy converted will remain small, and mate- �E0=E x pxe�ð Þ x 1 (12) rialization will not present a significant source of dissipa- � ½ � ¼ tion in practice. However, materialization of near-critical with! andh ffiffiffiz defined as above andB the Bernoulli 0 ð Þ 2k fields (E=E0 1) obtained in relativistic focusing can lead numbers. The ‘‘x coth x’’ found in the pair creation rate to the formation� of largeO 50 nm 3 spatial domains of [16,39,40] has been replaced due to the weighting of the electron-positron-photon plasmað withÞT 2 MeV, allow- phase space integral, resulting in a dependence on the ing experiments to test the strongly coupled’ regime of 2 magnitudeB . QED and provide an accessible analogy for the current Combining Eqs. (9) and (12) the materialization time is interest in quark-gluon plasma [46]. 1 �E =E In summary, we have studied the materialization time� uf !� eð 0 Þ � 0 : (13) of the electromagnetic field in view of pair production at ¼ E2 �E 1 h 0 � B 2;E high field intensity/energy density. We presented the field þ ð E Þ � ð Þ qffiffiffiffiffiffiffi 2 2 dependence of� and found that a field of order0:2E 0 is In Fig. 2, Eq. (13) is evaluated for B=E 10 � , 1, 10 . For sufficient for observable materialization. Our current study B*E the lifetime of the field is increased¼ over the pure electric case despite the augmented production rate evi- denced by the coth factor mentioned above. TABLE I. Materialization characteristics (yield rateW, relaxa- In Table I, we exhibit a few points of reference for a pure tion time�) for specific applied fields. 3 1 electric field, listing the particle creation rate from Eq. (3) E=E0 W �m � fs� � fs and the expected lifetime of the field given complete ½ � ½ � : 18 materialization from Eq. (7). At E=E 0:2 (4% of criti- 0.0628 12 275 10 0 0.13:102 10 8� 1:88 10 10 cal power intensity) and a time scale¼ of 10 15 s, for ex- � 0.28:234� 10 15 2800� ample, the materialization rate shows that 0.036% of the 0.4029:68� 10 19 1 field energy is converted and therefore approximately � 22 3 3 15:903 10 8:85 10 � 280 nC of electrons (positrons) created per�m � , a sig- � �

057901-3 54

BRIEF REPORTS PHYSICAL REVIEW D 79, 057901 (2009) relies on an adiabatically changing field configuration, as is ized to such more intense and shorter lived field configu- appropriate in the EHS context. This is consistent a poste- rations, because these calculations can be undertaken riori forE E 0 given the characteristic times for materi- within the same semiclassical approach. alization in! critical fields, which are 1000 times shorter This research was supported by the U.S. Department of than the typical intense pulse laser fields operating at 10 Energy Grant No. DE-FG02-04ER4131 and by the DFG– femtosecond scale. On the other hand, our results imply LMUexcellent grant. We thank for his generous hospitality that stronger fields E and closer investigations are nec- 0 Professor Dr. D. Habs, Director of the Cluster of essary for the much� shorter (attosecond range) field pulses Excellence in Laser Science—Munich Center for generated in relativistic focusing. The here introduced Advanced Photonics. concept of the field materialization time may be general-

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APPENDIX C

Dark Energy Simulacrum in Nonlinear Electrodynamics

Dark Energy Simulacrum in Nonlinear Electrodynamics Labun, L. and Rafelski, J. Physical Review D81 (2010) 065026 doi: 10.1103/PhysRevD.81.065026 arXiv:0811.4467 [hep-th]

Copyright (2010) by the American Physical Society Reprint permission granted under APS Copyright Policies: http://publish.aps.org/copyrightFAQ.html#thesis 56 PHYSICAL REVIEW D 81, 065026 (2010) Dark energy simulacrum in nonlinear electrodynamics

Lance Labun and Johann Rafelski Department of Physics, University of Arizona, Tucson, Arizona, 85721 USA, and Ludwig-Maximillians-Universita¨t Mu¨nchen, 85748 Garching, Germany (Received 19 March 2009; revised manuscript received 27 February 2010; published 25 March 2010) Quasiconstant external fields in nonlinear electromagnetism generate a global contribution proportional tog �� in the energy-momentum tensor, thus a simulacrum of dark energy. To provide a thorough understanding of the origin and strength of its effects, we undertake a complete theoretical and numerical study of the energy-momentum tensorT �� for nonlinear electromagnetism. The Euler-Heisenberg nonlinearity due to quantum fluctuations of spinor and scalar matter fields is considered and contrasted with the properties of classical nonlinear Born-Infeld electromagnetism. We address modifications of charged particle kinematics by strong background fields.

DOI: 10.1103/PhysRevD.81.065026 PACS numbers: 03.50.De, 04.40.Nr, 11.10.Lm, 12.20.Ds

I. INTRODUCTION TMax g 1F F�� F F � (2) �� ¼ ��4 �� Recent analysis has constrained the dark energy to the characteristics of a cosmological constant: equation of is manifestly traceless. Even in absence of external sources statew p=�6 1 and spatially homogeneous distribu- the Maxwell field equations are incomplete due to interac- � � tion [1,2]. This means that the dark energy is present in the tion with the electron-positron vacuum fluctuations which �� - energy-momentum tensor proportional tog as has often are present at the length scale� e @=m ec. At distances of - � been discussed in the context of vacuum energy [3]. comparable magnitude (� � e) these are vacuum polar- Proportionality tog �� does not exclude that dark energy ization effects which impact’ precision atomic physics ex- - originates in properties of ponderable fields and matter; in periments. For long distance (� � e) one obtains the the energy-momentum tensor nonlinear effective theory of the� photon studied in depth by Euler, Kockel, and Heisenberg, and Schwinger [5,6]. 2 � 4 This nonlinear Euler-Heisenberg (EH) theory of electro- T�� d xp gVeff; (1) ¼ p g �g�� magnetism is just one of many possible effective actions. � Z ffiffiffiffiffiffiffi Beyond the EH-QED framework, we can imagine writing ffiffiffiffiffiffiffi� �� a nonvanishing traceT � g T is a simulacrum of the down a more complete theory containing all effective ¼ �� dark energy, that is, a tangible but not necessarily complete interactions, reducing in the long wavelength limit at the analogy: the properties and consequences are the same, but classical level to Maxwell’s equations. Born-Infeld (BI) ultimately the generating mechanisms may differ between theory, designed to regulate point-particle-induced diver- � the observed dark energy and the modelT � . The possibil- gences [7], can be thought of as an effort to provide a more ity that the cosmological dark energy is consequence of a complete theory of electromagnetism, though it too is now (more complete) theory of ponderable matter and fields considered as an effective theory [8] arising from string motivates a deeper probe of well-controlled situations in theory. which the phenomenologically similar energy-momentum In nonlinear electromagnetism, the Maxwell energy- trace arises. momentum tensor Eq. (2) is modified by two quantities: As has often been noted, the small magnitude of the dark a dielectric function", which scales the contribution of the energy suggests a weakly broken symmetry as its source. Maxwell energy-momentum in the total, and the trace of � Hence the natural starting point for the present investiga- energy-momentum tensorT � , which is a new contribution. � tion is a theory with a traceless energy-momentum tensor. We have elsewhere observed thatT � arising in QED can in Extensions of this theory are analyzed for the generation of this regard be viewed as the modification of the vacuum an energy-momentum trace and by comparison with the energy by the presence of electromagnetic fields [9], and, starting theory the physics of the energy-momentum trace analogously, the connection of dark energy with a form of better understood. An effort in this direction has been made vacuum energy has been discussed before [3,4]. In the � in the context of the vacuum structure of quantum chro- external field framework of the EH theory, the traceT � modynamics (QCD) [4], and we show here that quantum is a global constant energy density proportional tog �� and electrodynamics (QED) has similar features which are thus serves as a simulacrum of a cosmological constant in more easily accessible. the ‘‘universe’’ spanned by the external field. In this work Our point of departure is therefore Maxwell electromag- we investigate the magnitude and some consequences of a � netism, whose energy-momentum tensor, ponderableT � -as-cosmological-constant, evaluate the di-

1550-7998=2010=81(6)=065026(16) 065026-1� 2010 The American Physical Society 57 LANCE LABUN AND JOHANN RAFELSKI PHYSICAL REVIEW D 81, 065026 (2010) electric function and compare the behavior of EH and BI ifications to the Maxwell energy-momentum tensor. In nonlinear theories. clarification of conflicting statements present in the litera- Now, as is well known, a theory with a traceless energy- ture, we begin our discussion of quantum electrodynamics momentum tensor is invariant under scale changes. in Sec. IV with a new derivation of the relationship be- Maxwell’s electromagnetism is the prime example: the tween the electron-positron condensate and the energy- energy-momentum tensor is traceless and indeed the clas- momentum trace based on the explicit origin of the trace sical theory of radiation is scale invariant. In a gauge in nonlinearity of the theory. Extending a technique of theory, the related conformal symmetry can be spontane- resummation of the action introduced by Mu¨ller, et al. ously broken, the study of which has a long and distin- [14], we then provide complete numerical evaluations of guished history, originating with the rise of QCD as the the condensate, energy-momentum trace and dielectric theory of strong interactions [10,11]. We will rederive here function for QED and spin-0 quantum electrodynamics. some results of the extensions of these studies to QED These evaluations display striking analytical features not [12,13]. before apparent in the Euler-Heisenberg functions. We � Scale invariance can be broken by the explicit appear- compare BI and QED contributions toT � and show that ance of a dimensionful quantity in the theory, such as the QED vacuum fluctuations remain dominant given the ex- mass of the electron in QED. In fact, any nonlinear elec- perimental constraints. tromagnetism requires a scale with the dimension of an In Sec. V of this report, we discuss the kinematics of electrical field eE0, in order to render the Lagrangian charged particles moving in external fields. The vacuum of dimensionally consistent. We express this scale in terms a nonlinear theory is studied as a ponderous medium with of a massM: nonlinear response. The Lorentz force is preserved, but the breaking of the superposition principle results in effective Mc2 2 potentials for charged particles moving in external fields eE ð Þ : 0 � @c that are not automatically obtained from the Lorentz force.

For the Euler-Heisenberg (EH) effective action the intro- II. ENERGY-MOMENTUM TENSOR OF duction ofM is a natural step to take as the nonlinearity is NONLINEAR ELECTROMAGNETISM of quantum origin andM m= p�, where� 1=137 is the usual fine structure constant.’ For the BI¼ theory it is a Setting from now on@ c 1, we can consider the ffiffiffiffi 2 ¼ ¼ matter of convenience to use mass rather than length as the effects ofE 0 : M =e massM or lengthl 1=M scale, scale, converting one into another using@. whereM can be¼ as large as a string theory scale¼ or as small If indeed BI theory is a weak-field limit of string theory as the mass of the electron. The consequences are best seen in which the nonlinearity is a consequence of high mass writing the effective action in the form quantum fluctuations [8] the appearance of@ would be appropriate, and the associated scale could be as large as S P � S 4 � S the Planck mass,M 1:2 10 19 GeV. It should be V eff M feff ; (3) Pl � � þ M4 M4 M! � noted that current experimental¼ � limits as well as EH non- � � !1 linearity probes a scale below 100 MeV, thus a string presented here as a function of the Lorentz scalar and related BI theory maybe quite removed from the present pseudoscalar experimental reality. We compare EH and BI theories mainly because their behavior is very different. S : 1F F�� 1 B2 E 2 ; (4a) In QED and BI, the presence of scale which breaks the ¼ 4 �� ¼ 2ð � Þ conformal symmetry is explicit, and the energy- P : 1 �� 4F�� F E B: (4b) momentum trace is not ‘‘anomalous,’’ unlike the case of ¼ ¼ � QCD. The scale of the nonlinearity is, as we shall show, the � As noted, classical, linear electromagnetism must consti- determining factor ofT � . This fact is the simple yet tute the limit of Eq. (3) for fields small as measured in units important and original theoretical observation presented � ofE 0. It should be noted that feff consists only of nonlinear in this paper. Having thus suggested the interconnection terms, specifically excluding linear terms such asS lnm=� of dark energy, conformal symmetry and the presence of which introduce another scale�. When such terms are scale in the theory, we leave issues specific to conformal admitted, the bar is left offf andV . symmetry to future work and here focus on the physics of eff eff � T� and its origins in nonlinearity of the theory. We derive in Sec. II the field energy-momentum tensor A. Dielectric function and Trace and explicitly connect its trace to nonlinearity of the elec- To understand the implications of the dimensioned scale tromagnetic theory. To compare relative magnitudes and we consider the explicit form of the energy-momentum suggest new constraints on a Born-Infeld-type completion tensor (1), separating the traceless Maxwell part. For a of electromagnetism, in Sec. III we evaluate the BI mod- general functionV S;P , we obtain effð Þ

065026-2 58 DARK ENERGY SIMULACRUM IN NONLINEAR... PHYSICAL REVIEW D 81, 065026 (2010) @V In view of Eqs. (5) and (6), we summarize T eff g S F F � �� @S �� ¼ �� ð � Þ T "T Max g 1T (10a) @V @V �� ¼ �� þ ��4 g V S eff P eff : (5) � �� eff S P @Veff � dfeff � � @ � @ " ;T� T M : (10b) � � �� @S � ¼� dM Comparison with Eq. (2) shows the energy-momentum Here" is the dielectric function,T Max the Maxwell energy- tensor of Maxwell’s electromagnetism is modified by a �� momentum tensor, andT the energy-momentum trace. dielectric function, @V =@S, to be discussed below. eff Interestingly,T=4 �=2 provides a dark energy or Using Eq. (3) we simplify� the second term to Einstein-like cosmological$ constant, while the traceless � part is the same as in Maxwell theory, up to the multi- S @Veff P @Veff 1 @feff Veff M ; (6) plicative dielectric function. � @S � @P ¼ 4 @M � � Equation (10b) though derived here for the case of This form is very useful because it provides a simple means electromagnetism has a much wider domain of validity. of calculating the trace directly from the effective action. The implications of the separation in Eq. (10) have been The importance of Eq. (6) lies in its distillation of the previously noted in context of the photon propagation physical source of conformal symmetry breaking in any effects [15,16], butT was not given in the form nonlinear theory of electromagnetism: Eq. (10b), nor has in any form the traceT been computed. Terms linear in the invariantS cannot contribute to the We shall below demonstrate how the concomitant identi- right side of Eq. (6) since they cancel explicitly on the left ties Eq. (10b) provide new physical insight in understand- side of Eq. (6). Such contributions must therefore be ing the quantum effects and their connection to omitted fromV eff in the study of energy-momentum trace, nonlinearity of the action. � and hence we have introduced the barred feff to denote the The alternative and equivalent representation, often used nonlinear components of the effective potential. Letting in the electromagnetism of nonlinear media (for example 1 see Sec. 8 of [17]), Veffð Þ denote the remaining linear terms in the Lagrangian, we have the decomposition T�� H ��F� g ��L (11) ¼ � � 1 4 V V ð Þ M f� : (7) is using the displacement tensor eff ¼ eff þ eff S P @L @L @L The lowest power in , is 2 due to preservation of both H�� F�� F~��: (12) parity and charge conjugation symmetry, which, respec- ¼ @F�� ¼ @S þ @P tively, require an action even under parity transformations and even in the couplinge, hence even in the field strengths The trace is now distributed into several components (already true of any Lorentz scalar). A nonzero imaginary �� ~ ~ ~ ~ T H F�� 4L E D B H 4L: (13) part of the action entails breaking of time reversal symme- ¼ � ¼ � þ � � try, though the traceT will be small under laboratory The origins and properties ofT are obscured by the conditions. These symmetry arguments imply that for field constitutive relationH �� F�� . The expression Eq. (10) strengths below the critical scaleE m 2, the energy- evidences the departureð fromÞ the classical theory more momentum trace must be at least 4th� order in the fields. clearly in the context we consider. Solving the partial differential equation @V @V B. Stress-energy density V S eff P eff 0 (8) eff � @S � @P ¼ Some further notable properties of the stress-energy density of the nonlinear electromagnetic (EM) field will displays one obvious class of nonlinear Lagrangians that be collected here. The energy density in the frame of the have traceless energy-momentum tensors, namely, metric, i.e. the quantity entering Einstein’s equations, is

S n T00 " E2 B 2 1T: S þ1 ¼ 2ð þ Þ þ 4 Veff an P : (9) ¼ n � � ij ¼�1X The EM-stressesT have the same structure as in the Maxwell theory, as is already evident from the format of Comparison with Eq. (3) reveals the reason: such Eq. (16): Lagrangians are conformal. Since this class is nonperturbative in at least one of the ﬁeld invariants and we are ij ij 1T T "T Max � ij (14) interested in having an energy-momentum trace, we ex- ¼ � 4 T ij clude theories of the form in Eq. (9) as well any other The trace acts to compensate the forcesT Max tearing the nonperturbative actions that may satisfy Eq. (8), despite ﬁeld sources apart in Maxwell electromagnetism. For this their inherent interest. reason, for example, in BI theory theT ij vanishes allowing

065026-3 59 LANCE LABUN AND JOHANN RAFELSKI PHYSICAL REVIEW D 81, 065026 (2010) a stable charged particle without material stresses. A suffi- arising in consideration of the radial electric fieldE r of a cient condition for this to be true is that the point particle point chargeq: solution satisfies lim r3T00 0[ 18]. r 0 1 q Because the energy-momentum! ¼ tensor is conserved we 3 2 U d x Er ; forE r 2 haveT @T =@x� 0, which is a covariant relation ¼ 2 ! 1 ¼ r ��;� � �� ¼ Z true in any frame. A differential conservation law leads to To remedy this, Born and Infeld took inspiration from an integral conservation law by integration over the ob- special relativity, considering the action [note that we server’s hypersurface: follow the modern convention, opposite in sign to the 2 �� original paper [7], and also, recall remarks about the scale 4 @T 3 �� 3 �� d x � 0; or d ��T d ��T ; M above Eq. (3)]: 1 @x ¼ 1 ¼ 2 Z Z Z (15) BI 4 S 4 P 4 2 Veffð Þ M 1 1 2 =M =M (19) i.e. the energy-momentum flow through surface 1 is the ¼ ð � qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ � ð Þ Þ same as later through surface 2. It is common to choose an 4 3 3 M p g p h ; (20) observer at rest in laboratory so thatd �� u �d x, with ¼ ð � � � Þ u� 1;0;0;0 . ¼ ffiffiffiffiffiffiffi ffiffiffiffiffiffiffi For¼ ð nearlyÞ homogeneous fields we can omit the 3- F�� h deth ��; h�� g �� ; (21) volume and consider a conserved four-momentum density ¼ ¼ þ M2 of the EM field where the particular combination ofS,P terms derives � �� 2 2 ~ ~ from the extension of the space-time metric with the anti- pMax u �TMax E B =2; E B ¼ ! ðð þ Þ � Þ symmetric field tensor,F ��. In the weak field (infinite BI finding the well-known result for the rest energy and mass) limit Maxwell’s theory indeed arises,V ð Þ S. Poynting vector of the classical field. This result is easily For the Born-Infeld case the dielectric function! is � generalized to the nonlinear electromagnetism: BI @Vð Þ E2 B 2 T " eff 1 2S=M 4 P =M4 2 1=2; (22) p� " þ ;"E~ B~ ; (16) BI ¼ � @S ¼ ½ þ � ð Þ �� ¼ � 2 þ 4 � � which exhibits a formal analogy to the�-factor familiar showing the appropriateness of calling" the dielectric from special relativity, though with two different limits as function, since it plays the role of the dielectric constant S orP respectively approach the limiting valueM 4 (see ~ ~ E " D when considering electric charge in vacuum. For Fig. 1, left). The dielectric function goes over from sup- Maxwell¼ electromagnetism" 1 andT 0. ¼ ¼ pression ("<1) to augmentation (" >1), when the mag- It is an elementary exercise to show that in the Maxwell netic component of the field becomes subdominant, which limit the proper energy density, or its ‘‘mass density,’’ is corresponds to crossing the line2S P 2 from the lower ¼ E2 B 2 2 right. Max � ~ ~ 2 p� pMax ð þ Þ E B (17a) As presented, Eq. (19) is in the form required by Eq. (3). ¼ 4 � ð � Þ S T 2 2 2 However, the BI action contains no terms linear in ; is E B 2 2 2 ð � Þ E~ B~ S P : (17b) identically M dV eff=dM . We obtain for the BI energy- ¼ 4 þ ð � Þ ¼ þ momentum� traceð using theÞ relation Eq. (6) Generalizing to nonlinear EM theory, we find the local T BI 4M 4 " 1 S=M 41 1 ; mass density of the field to be ð Þ ¼ ð BIð þ Þ � Þ 1 2S=M 4 S=M4 2 u p p� S2 P 2 "2 T=4 2: (18) 4M 4 þ þ ð Þ 1 (23) f � � ¼ ð þ Þ þ ð Þ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2S=M 4 P =M4 2 � qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � þ � ð Þ � T=4 �=2 provides thus both a ‘‘dark energy’’ and a $ which in the latter form is manifestly positive-definite, just ‘‘mass density’’ of the electromagnetic field. like the cosmological constant. For small fields we expand the second form in Eq. (23) to obtain III. BORN-INFELD ELECTROMAGNETISM 2 S2 P 2 T BI : (24) As demonstrated in the preceding discussion, an intrinsi- ð Þ 4 þ ! M 1 2S=M 4 cally nonlinear theory of electromagnetism (in most cases) þ entails an energy-momentum trace, and we begin by study- In Fig. 1 we show the dielectricpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi function and energy- ing a nonquantum example of a nonlinear alternative to momentum trace for strengths up to the maximum field Maxwellian theory. Historically, Born-Infeld electromag- strength. The functional behavior of both is smooth, though netism was introduced in order to solve the infinite self- aS,P functional asymmetry develops at large values of energy (and self-stress) problem of a pointlike electron the fields.

065026-4 60 DARK ENERGY SIMULACRUM IN NONLINEAR... PHYSICAL REVIEW D 81, 065026 (2010)

�� FIG. 1 (color online). The departure of the Born-Infeld energy-momentum tensor from that of the MaxwellT Max: At left the dielectric function, Eq. (22) and at right the trace, Eq. (23). Field strength invariants and energy densities are in units ofM 4.

In contrast to the classical theory analyzed here, we note sign corresponds to the difference in sign of vacuum fluc- a recent report suggesting that the quantized BI theory as tuations between bosons and fermions. studied on the lattice may be conformally symmetric [19]. A. The condensate c� c and traceT h i IV. EULER-HEISENBERG ELECTROMAGNETISM Equation (10b) provides a direct means of calculating the energy-momentum trace, but its connection to vacuum The Euler-Heisenberg effective action is well known: structure in QED and its deformation by the applied fields m2s which induce the EH nonlinearity is encoded in f 1 dse� Veff 2 3 1 eas cot eas ebs coth ebs m dVeff=dm . Consider the Feynman boundary condition ¼ 0 � 8� s ð � ð Þ ð ÞÞ ð Þ Z þ Green’s function of the fluctuating matter field in presence (25) of the electromagnetic field, which determines the vacuum for Dirac fermions, and fluctuations,

2 @V x; m dse m s eff 0 s 1 � m ð Þ im lim tr SF x �; x �; m S Fð Þ Veff 2 3 eas csc eas ebscsch ebs 1 @m ¼ � 0 ½ ð þ � Þ � � ¼ 0 � 16� s ð ð Þ ð Þ� Þ ! Z þ (28) (26) 0 Here� is a timelike vector, andS ð Þ is the free field for charged scalars, in which F Feynman Green’s function. In general � : S2 P 2 S : S2 P 2 S SF x; x0 i T c x0 c x : (29) a ; andb : ð Þ ¼ � h ð ð Þ ð ÞÞi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ � ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (27) We rewrite the right side of Eq. (28) using the elementary form of Wick’s decomposition theorem The characteristic strength of fluctuations in the matter field is made explicit in the appearance ofm which is the T c x0 c� x : c x0 c� x : 0 T c x0 c� x 0 ; mass of the matter particle which has been integrated out; ð ð Þ ð ÞÞ ¼ ð Þ ð Þ þh j ð ð Þ ð ÞÞj i in QED it is the mass of the electron. where the normal ordering is with respect to the ‘‘no-field’’ vacuum. Taking the expectation value of this relation at a For vanishingP we havea S S andb ! j j� ! single space-time point in the ‘‘with-field’’ vacuum we S S . Thus whenE 0 we finda p0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi,b B and ji j jþ ¼ ! !j j find pwhenffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB 0 we finda E ,b 0. In this sense,b plays the role¼ of the generalized!j j ! (Lorentz invariant) magnetic :c c� : T c x c� x 0 T c x c� x 0 h i ¼ hj ð ð Þ ð ÞÞji � h j ð ð Þ ð ÞÞj i field anda that of the generalized electric field. The iS x; x iS0 x; x (30) constant subtraction 1 removes the (divergent) zero point ¼ Fð Þ � Fð Þ energy of free electrons� and positrons. The difference in with the same�-limit as in Eq. ( 28) implied for equal normalization reflects the doubling of the number of de- propagator arguments. Usually the trace is implied by grees of freedom for spin-1=2 particles, and the overall commuting the fields and the normal ordering symbols

065026-5 61 LANCE LABUN AND JOHANN RAFELSKI PHYSICAL REVIEW D 81, 065026 (2010) omitted, m dVeff=dm renders the lowest order contribution ð 1 Þ m dVð Þ=dm finite. The condensate and energy- m :c c� : m c� c ; ð eff Þ h i ! � h i momentum trace are thereby independent of renormaliza- hiding the important operational definition Eq. (30) of what tion procedure (as they should be), and obey the nontrivial we now recognize as the fermionic condensate. Thus, we relationship expressed by Eq. (32). The finiteness ofT for see that the condensate derives from the difference of fermions is also discussed at length by Adler et al[ 12]. normal ordering in the no-field (also called perturbative) Separation of the term linear inS in the Fermi case vacuum and the with-field vacuum. Furthermore, one must shows keep in mind that the subtraction of the unperturbed vac- 1 2 2 2 dVð Þ e b a ds 2 2� uum term follows directly from application of the rules of eff 1 m s S m 2 � e� : (34) QED and is not a consequence of arbitrary removal of zero- dm ¼ � 8� 3 0 s ¼ 3� point energy. For this reason our discussion has no bearing Z on the zero-point energy of quantum field theory or its Thus for the EH effective action, the decomposition in gravitational coupling. Eq. (7) becomes Equations (29), (28), and (30) then combine with the dVf dV�f 2� result m eff m eff S (35) dm ¼ dm þ 3� dV m eff m c� c ; (31) dm ¼ � h i and in turn with emphasis on properties of the vacuum: a known and widely used relation in nonperturbative QCD. 2� We will evaluate it for the case of electrons in the presence T f S m c� c : (36) ¼ 3� h iþ h i of external electromagnetic fields. We undertook the derivation of the Fermi condensate in Equation (36) corresponds to Eq. (2.17) in [12]. It is of the preceding section in order to emphasize that the con- importance to note that there is considerable cancellation densate is not in general the energy-momentum trace, between the two terms on the right-hand side [4]. though there is a relationship. Using Eqs. (7), (10b), and The essential relation Eq. (36) must be preserved at any (31) we find number of loops in the effective action in its suitable 1 generalization. In particular, if the condensate were eval- dVð Þ T m eff m c� c : (32) uated to two loops, the coefficient of the first term must ¼ dm þ h i become the two-loop� function, i.e. 1 If and only ifV ð Þ is just the action of classical electro- eff 2� 2� �2 magnetism, the first term on the right-hand side, linear in � � 2 ...: (37) S, vanishes. 3� ! ð Þ¼ 3� þ 2� þ However, the coupling to a fluctuating quantum field Our result Eq. (36) is not obvious if one evaluates the complicates the issue significantly in that it produces a energy-momentum trace after renormalization has been S contribution linear in with a coefficient which is a carried out. The logarithmically divergent term inV is function ofm and renormalization scale. It is the logarith- eff mic divergence of Eqs. (25) and (26) which in the limit 2 2 2 1 e b a ds m2s � 2 1 Vð Þ 1 e S ln m � S ð Þ eff 2 � � � 0 is also proportional to and thus appears inV eff . ¼ � 8� 3 0 � s ¼ 3� ð Þ ! Z þ The standard procedure is to absorb the divergence into the (38) definition of the charge in the process of charge renormalization. Equations (25) and (26) assume a cutoff regulator in which� 1=M 2, some large mass or momentum ex- which of course also introduces a scale. Alternatively one traneous to¼ QED. Before presenting the EH action, this can use dimensional regularization, the spinor effective term is absorbed in the process of charge renormalization. potential can be written To restore its contribution one must realize the mass de-

2 pendence of charge renormalization. The relation of Eq. dse m s f 1 � (38) to the QED� function [ 12] demonstrates why use of Veff 2 3 � 1 eas cot eas ebs coth ebs ¼ 0 8� s � ð � ð Þ ð ÞÞ the renormalizedV with the incorrect identificationT Z eff ¼ (33) mdVeff=dm leads to the correct result as shown in Eq. (36), just as it was developed for QCD [4,10,11]. and similar for the charged scalar case. For any finite�, Eq. ( 33) is finite and can be differentiated with respect tom, and hence the quantity B. Properties ofT in QED f;s m dVeff=dm is finite, allowing the consideration of a We obtain the explicit form of the condensate in external vanishingð �.Þ In particular, the differentiation fields combining Eq. (31) and (25)

065026-6 62 DARK ENERGY SIMULACRUM IN NONLINEAR... PHYSICAL REVIEW D 81, 065026 (2010) dse m2s 1 1 � 2 1 � PV i�� x : m c c m 2 2 � h i ¼ 0 4� s x i� ¼ x þ ð Þ Z � eas cot eas ebs coth ebs 1 : (39) As befits its role contributing to the proper mass of the � ð ð Þ ð Þ � Þ nonlinear electromagnetic field [recall Eq. (18)], The condensate vanishes in the absence of field, as it eE 1 1 should, sincex cotx 1 andx cothx 1 forx 0. The ImT f m2 e n�E0 =E (45a) ! ! ! 2 �ðð Þ Þ term quadratic in the fields has been discussed above, it ¼ � 4� n 1 n X¼ must be subtracted to arrive at the integral representation of 4 m � the energy-momentum trace: ln 1 e � ; (45b) ¼ 4�� ð � Þ dse m2s T f 2 1 � is manifestly negative and strongly suppressed for field m 2 2 ¼ � 0 4� s strengths less than0:1E . This is consistent with direct Z 0 2 differentiation of the positive imaginary part of the action e 2 2 2 eascot eas ebscoth ebs 1 b a s : Vf evaluated by Schwinger � � ð Þ ð Þ� � 3 ð � Þ � eff 2 eE 1 1 (40) f n�E0 =E ImV eff ð Þ3 2 e�ðð Þ Þ: ¼ 8� n 1 n To study the integrals Eq. (39) and (40), we introduce a X¼ transformation that will be helpful in dealing with the Observe thatm c� c in external magnetic fields is nega- nonanalyticities generated by the electric field. The de- tive, as is manifesth i in Eq. (41), while the energy- tailed calculations are carried out in the Appendix, and momentum trace, Eq. (42) is positive. With the perspective we simply state here their results. The integral representa- that the energy-momentum trace represents the energy tions obtained along the way have better convergence arising from deformation of the vacuum, a negative value properties than Eq. (39) or Eq. (40), particularly at strong would imply that the vacuum state is unstable, for example, fieldsB,E E . to the spontaneous generation of strong magnetic fields if � 0 In a magnetic background field, the condensate can be T were identicallym c� c . written The energy-momentumh tracei in general combinations of electric and magnetic can also be cast Eq. (A9) reminiscent m4 ln 1 e �0s � 1 � of our prior study of the special cases of electric and m c c 2 ð �2 Þ ds; (41) � h i ¼ � 2� �0 0 s 1 magnetic fields. Usingb in the definition of� anda in Z þ 0 2 2 �, the numerically useful representation is where� 0 : �m =eB �= B=E 0 , withE 0 m =e. The energy-momentum¼ trace¼ isð obtainedÞ by removal¼ of the m4 T QED I I (46a) leading term quadratic in the field [recall Eq. (36)], with ð Þ ¼ 2�2 ð a þ bÞ 2 sk� the result that dss 1 e �k� �k� I 1 � coth ; (46b) a 2 k� � � m4 s2 ln 1 e �0s ¼ 0 1 s i� k 1 0 0 T f 1 ð � � Þ ds (42) Z � � X¼ 2 2 2 sk�0 ¼ � 2� �0 0 1 s 1 dss 1 e� �k�0 �k�0 Z þ Ib 2 coth : (46c) ¼ 0 1 s k 1 k�0 � � is manifestly positive definite. Z þ X¼ For the electric field, the poles are resummed into a A further resummation (see the Appendix) displays a more logarithmic winding point, with the result statistical form of the integrands: 4 �s s 1 ln 1 x i� m 1 ln 1 e � 1 2 1�xþi� m c� c ds (43) I b þ ð þ � Þ ds (47a) 2 ð �2 Þ a �H � h i ¼ 2� � 0 1 s i� ¼ � 0 e � 1 Z � � Xn;� Z � s arctans where� : �= E=E0 . The trace in the pure electric back- 1 Ib a �H ds (47b) ground is ¼ ð Þ ¼ 0 e�0 �0 1 Xn;� Z � m4 s2 ln 1 e �s where� 1 andn 0;1;2... in accordance with the T f 1 � ¼� ¼ 2 ð �2 Þ ds: (44) Landau levels apparent in the quasi-Hamiltonians ¼ 2� � 0 1 s i� Z � � Eqs. (43) and (44) could equivalently be obtained by m2H m 2s 2n 1 � eb (48a) � ¼ þð þ � Þ takingB iE in the respective magnetic expressions. The 2 2 ! 2 m H m s 2n 1 � ea: (48b) condensate behaves as eE for small fields, but the poles �0 ¼ þð þ � Þ displayed in Eq. (A7) giveð Þ the condensate and the energy- momentum trace a nonzero imaginary part, which can be The slower convergence ofI a (Ib) for b=a 1(a=b 1) evaluated from Eq. (43) or (44) recalling the identity means Eq. (47) is not as advantageous� as Eq. (�46) for

065026-7 63 LANCE LABUN AND JOHANN RAFELSKI PHYSICAL REVIEW D 81, 065026 (2010) 10000 µ 100 Tµ (Ε) perturb µ 1 Tµ (E) µ 0.01 − Tµ (E) µ ] 4 0.0001 − Im Tµ (Ε) [m 1e-06

1e-08

1e-10

1e-12 0.001 0.01 0.1 1 10 100 1000

E / E0

� 4 FIG. 2 (color online). The traceT � normalized by the electron massm e for spin-1=2 matter fields in magnetic (left panel) and electric (right panel) fields, evaluated perturbatively (dashed-dotted line) from the first two terms in Eq. (50) and exactly (solid line) from Eqs. (42) and (44). At left, the condensate in magnetic-only backgrounds is included for comparison of magnitudes, and at right, the imaginary part present for electric fields is included. The trace changes sign from positive to negative atE 9E 0 and for higher � ¼ fields the negative ( T � ) is plotted. �

numerical work. These expressions clarify the electric and terms for the condensate and energy-momentum trace are magnetic field limits and thereby the correspondence be- m4 E2 m4 E4 tween Figs. 2 and 3 below when one also recallsz cothz m c� c ... (49) ! 2 2 2 4 1 forz 0. � h i � 12� E0 � 90� E0 þ ! m4 E4 4m4 E6 C. Numerical evaluation of QEDT T f ... (50) � 90�2 E4 � 315�2 E6 þ We evaluate numerically the condensate and the energy- 0 0 momentum trace for arbitrary constant, homogeneous whereE 2 B 2 orE 2 E2 for magnetic or electric ¼ ¼ � background electromagnetic field. We consider first a fields, respectively. magnetic-only background field in Eq. (39), a form which The condensate and energy-momentum trace for more is easily integrated numerically. The behavior is displayed general field configurations are evaluated using the rapidly in Fig. 2 as the solid upper (red) line. Noting thatf x convergent sums in Eq. (A9), and the results displayed in x cothx 1 =x 2 is essentially constant forx<1, weð Þ re- ¼ ðcover the� Þ quadratic dominance m c� c eB 2 for small fields. Figure 2 confirms� theh smalli field/ ð (mass-Þ dominant) quadratic behavior and large field (field- dominant) linear behavior [20]. The results of numerically integrating (43) also appear in Fig. 2, but since the result is opposite in sign to the magnetic field case, we show the negative of the result as the lower (black) curve. Equations (44) and (42) are plotted on the right in Fig. 2, and interestingly, we see that atE 9E 0 T changes sign from positive at low fields to negative¼ at high fields. This result is not apparent in the perturbative expansion, but can be understood from the Cauchy-Riemann equations in view of the rapidly changing imaginary part. Having the appearance of a� phase flip, the feature may be related to the rapid dissolution (attosecond timescale) of fields at magnitudes surpassingE [21]. The methods developed 0 FIG. 3 (color online). The spinor energy-momentum trace for here are not suited to the dynamics implied by such fields generalE;B fields, parametrized by the Lorentz invariants,S, and the processes used to obtain such field strengths. For 2 P defined below Eq. (3). As onlyP appears inV eff, we plot only comparison, we also plot the weak field expansions, de- positiveP . Although the figure has a logarithmic scale, the trace rived from the original EH expression with the power-law crosses zero at the dotted line, going from positive to negative for semiconvergent expansions of cot and coth. The first two very large electric fieldsS 1, as seen also in Fig. 2. ��

065026-8 64 DARK ENERGY SIMULACRUM IN NONLINEAR... PHYSICAL REVIEW D 81, 065026 (2010) Fig. 3. An examination of the condensate for arbitrary field the explicit form of the energy-momentum tensor, Eq. (5), configurations, which does include the term linear inS (but remains valid, we need only manipulate the action of scalar is not displayed here), shows nontrivial features near the electrodynamics. lineS P . This is reflecting on the expectation [ 22], of The alternating sign in the meromorphic expansions of the influence¼ of zero modes, known to be present when the the csc and csch functions (see the Appendix) leads upon field is self-dual, i.e.S P . Comparison of the appear- partial integration to the opposite sign, or opposite ‘‘statis- ¼ �s ance of Fig. 3 with the BI result, Fig. 1 reveals a profound tics’’ in the logarithm ln 1 e � , as was already re- difference in these results, with the vacuum fluctuation marked upon by [14]. Thus,ð � we haveÞ for magnetic-only effective action being more ‘‘edgy,’’ and suggesting the background fields possibility of a more singular behavior in the full multiloop 4 �0s strong field case. 2 m 1 ln 1 e � m �� 2 ð þ 2 Þ ds (57) The weak field expansions in general electromagnetic h i ¼ 4� � 0 1 s 0 Z backgrounds give þ and for electric-only fields m4 m4 � S S2 P 2 4 �s m c c 2 2 4 7 ... (51) 2 m 1 ln 1 e � � h i � 6� � 90� ð þ Þ þ m �� 2 ð þ2 Þ ds: (58) h i ¼ 4� � 0 1 s i� Z � � m4 4m4 T f 4S2 7P 2 S 8S2 13P 2 ... Results of numerically integrating Eqs. (57) and (58) ap- � 90�2 ð þ Þ � 315�2 ð þ Þ þ pear in Fig. 4. As before, the condensate appears with (52) opposite signs when comparing the electric and magnetic S P backgrounds, and the pole structure remains consistent for using , normalized to the natural field strength,E 0. We electric and magnetic fields, irrespective of particle type. can compare the perturbative EH result (52) with the Born- T The imaginary part induced by the electric background Infeld weak field Eq. ( 23), but because the coefficients similarly becomes ofS 2 andP 2 do not match in QED, we obtain two values s n for the BI-type limiting mass, in obvious notation: dV eE 1 1 eff 2 n�E0 =E Imm m 2 ð� Þ e�ðð Þ Þ (59a) 45 1=4 dm ¼ 8� n 1 n MS m 15:16m; (53a) X¼ ¼ 16�2 ¼ m4 � � � ln 1 e � ; (59b) 45 1=4 ¼ 8�� ð þ � Þ orM P m 13:18m: (53b) ¼ 28�2 ¼ � � which is again negative in continued agreement with the role ofT in the proper mass-energy of the nonlinear electromagnetic field. D.T of scalar QED Finally, the energy-momentum trace generated by the scalar field quantum fluctuations is The Euler-Heisenberg-Schwinger calculation of the effective action is easily extended to the case of a charged m4 s2 ln 1 e �0s T s 1 � scalar field. The spin-0 particle has no Pauli spin coupling 2 ð þ 2 Þ ds; (60) �� ¼ 4� �0 0 1 s ��F in the Hamiltonian, and the proper time integral is Z þ evaluated for the Klein-Gordon equation with covariant in the magnetic background, and derivative, m4 s2 ln 1 e �s T s 1 � 2 2 2 ð þ2 Þ ds (61) D m � 0; D � @ � ieA�; (54) ¼ � 4� � 0 1 s i� ð þ Þ ¼ � � Z � � leading to Eq. (26) displayed above. The condensate takes in the electric background. Similarly positive for all but the the slightly different form,m 2 �� . Analogous manipu- h �i highest electric fields, the scalar energy-momentum trace lation of the proper time representations as above (see is exhibited in Fig. 4 on right. The effect of the fermionic Sec. IVA) leads to an identity similar to (31): statistics is apparent in the shift of the zero crossing toE ’ s 20:2E0. In the figure, we also compare the weak field 2 @Veff m �� m : (55) expansions, h i ¼ @m 4 E2 4 E4 In turn, we have 2 m 7m m �� 2 2 2 4 ... (62) � h i � 48� E0 � 1440� E0 þ T s S m 2 �� ; (56) ¼ 6� � h �i 7m4 E4 31m4 E6 which displays for the scalar field the large cancellation T s 2 4 2 6 ... (63) between the photon and matter condensates. Then, since � 1440� E0 � 5040� E0 þ

065026-9 65 LANCE LABUN AND JOHANN RAFELSKI PHYSICAL REVIEW D 81, 065026 (2010) 10000 10000 µ µ 100 Tµ (B) perturb 100 Tµ (E) perturb µ µ 1 Tµ (B) 1 Tµ (E) µ 2 ∗ − T (E) 0.01 m <φ φ(B)> 0.01 µ µ ] ] 4 4 − Im T (E) 0.0001 0.0001 µ [m [m 1e-06 1e-06

1e-08 1e-08

1e-10 1e-10

0.001 0.01 0.1 1 10 100 1000 0.001 0.01 0.1 1 10 100 1000

B / E0 E / E0

4 FIG. 4 (color online). For spinless particle fluctuations, we display the energy-momentum trace in units ofm e for both magnetic-only (left) and electric-only (right) backgrounds. Again, the condensate is displayed for comparison in the magnetic case, and the imaginary part of the trace present in the electric background. Note the change in sign of the trace is now atE 5:5E . ¼ 0 whereE 2 B 2 orE 2 E2 for magnetic or electric Eq. (47) can be obtained from that expression by changing fields, respectively.¼ ¼ � to the fermionic sign in the denominator and omitting The results for general electric and magnetic back- the spin sum andþ � term in the quasi-Hamiltonians grounds are Eq. (48); the spectral functions m4 1 1 x i� T s I I ; (64a) s ln � þ ands arctans 2 a b þ 2 1 x i� � ¼ � 4� ð þ Þ � þ � � 2 k sk� dss 1 1 e� �k� �k� remain unchanged. The weak field expansions for general I 1 ð� Þ csch ; a 2 electromagnetic backgrounds are ¼ 0 1 s i� k 1 k� �0 �0 Z � � X¼ (64b) m4 m4 2 S S2 P 2 m �� 2 2 7 ... (65) 2 k sk�0 h i � 24� � 360� ð þ Þ þ 1 dss 1 1 e� �k�0 �k�0 Ib 2 ð� Þ csch : (64c) ¼ 0 1 s k 1 k�0 � � 4 4 Z þ X¼ m m T s 7S2 P 2 S 31S2 11P 2 ... which is plotted in Fig. 5. The statistical form analogous to � 360�2 ð þ Þ � 630�2 ð þ Þ (66)

usingS,P normalized to the natural field strength,E 0. Again, we compare with the Born-Infeld weak-field energy-momentum trace [see Eq. (23)], and as above, the coefficients ofS 2 andP 2 are not the same. For the scalar quantum theory then, the two corresponding values for the BI-type limiting mass are 45 1=4 MS 2 m 18:64m; (67a) ¼ �7� � ¼ 45 1=4 andM P 2 m 30:32m: (67b) ¼ �� ¼

E. Euler-Heisenberg dielectric function

FIG. 5 (color online). The scalar energy-momentum trace for As we will discuss below, the kinematical and gravita- generalE;B fields, Eq. ( 64), parametrized by the Lorentz in- tional effects of a trace contribution to the electromagnetic variants. The dotted vertical line indicates the change in sign of energy-momentum differ strikingly from the classical the trace to negative values for very large electric fields. The Maxwell energy-momentum Eq. (2). Although this fact transition is present up to arbitrary values ofP as in Fig. 3. should make experimental verification of the presence of

065026-10 66 DARK ENERGY SIMULACRUM IN NONLINEAR... PHYSICAL REVIEW D 81, 065026 (2010) the trace term easy in principle, the relative strength of the the proper time variable persisting in the absence of the classical contribution cannot be ignored in the study of real m-differentiation makes for simpler ‘‘spectral’’ functions physical systems. Thus, we complete the analysis of the ln 1 s 2 i� and ln 1 s 2 energy-momentum tensor of Euler-Heisenberg electro- ð � þ Þ ð þ Þ magnetism with evaluation of the dielectric function" for the electric and magneticlike integralsK andK . S ¼ a b @Veff=@ . The weak ﬁeld expansion of the dielectric function can � The EH actions are corrections to the classical S, so � be obtained by straightforward differentiation of the ex- the total dielectric functions have the form pansion of the effective action, giving

f 2 4 f @Veff f � e 2� e " 1 : �" ; (68a) �"f 6 8S 24S2 13P 2 ... (72) � ¼� @S ¼ � 90� m4 þ 315� m8 ð þ Þþ @Vs "s 1 eff : �" s; (68b) For completeness, we exhibit the dielectric functions for � ¼� @S ¼ magnetic-only requiring differentiation of the expressions Eq. (25) and 2 k� s S 2� s s 2 1 e 0 (26), via the partial differentials @a=@ @=@a and �"f B 1 ds ð þ Þ � (73) S ð Þ 2 2 2 2 @b=@ @=@b, where ð Þ ¼ � � 0 s 1 k 1 k � ð Þ Z ð þ Þ X¼ @a a @b b � ; and : and electric-only @S ¼ a2 b 2 @S ¼ a2 b 2 þ þ 2 k�s f 2� 1 s s 2 1 e� Considering first the fermionic case Eq. (25), we find �" E ds ð2 � Þ2 2 2 : (74) ð Þ ¼ � � 0 s 1 k � Z ð � Þ k 1 1 ds eas cot eas ebs coth ebs X¼ f 1 2 2 �" 2 3 ð 2 Þ 2 ð Þ backgrounds, recalling� 0 �m =eB and� �m =eE ¼ 8� 0 s a b ! ! Z þ in the respective limits. cothebs ebscsch 2ebs coteas � ð � � For the scalar case Eq. (25), 2 2 2 m2s m2s eascsc eas b a s e� ; (69) s 1 e� ds eas csc eas ebscsch ebs þ � 3 ð þ Þ Þ �" 2 3 ð 2 Þ 2 ð Þ ¼ � 0 16� s a b Z þ for which renormalization only requires subtraction of the b a logarithmic divergence, since the zero-field constant is ebs cothebs eas coteas þ s2 ; (75) differentiated away. Obtaining the meromorphic expansion � � � � 3 � of the integrand by differentiating the Sitaramachandrarao again renormalized by subtraction of the logarithmic di- identity Eq. (A8), used above in Eq. (46), we have the vergence. The identity used above in Eq. (64), provides the numerically more convenient representation numerically more convenient representation

4 4 e ab 2 e ab 2 �"f 1 se m s K K ds; (70a) s 1 m s 2 2 2 � a b �" 2 2 2 dsse� Ka K b (76a) ¼ 2� a b 0 ð þ Þ ¼ � 4� a b 0 ð þ Þ þ Z 2 þ Z 1 k� coth k�b=a b=a csch k�b=a 2 2 1 k k�b k� Ka : a ð Þ ð Þ ð Þ K : a 1 csch ¼ eas 2 k 2�2 2 � eas 2 k 2�2 a a 2 2 2 2 k 1� � ¼ k 1ð� Þ � �� eas k � X¼ ðð Þ � Þ ð Þ � X¼ ðð Þ � Þ (70b) b=a coth k�b=a ð Þ ð Þ (76b) 2 � eas 2 k 2�2 2 1 k� coth k�a=b a=b csch k�a=b � Kb : b ð Þ ð Þ ð Þ : ð Þ � 2 2 2 2 2 2 2 1 k�a k� ¼ � k 1 ebs k � þ ebs k � 2 k �ðð Þ þ Þ ð Þ þ � Kb : b 1 csch X¼ b ebs 2 k 2�2 2 (70c) ¼ � k 1ð� Þ � �� X¼ ðð Þ þ Þ a=b coth k�a=b This form also provides the imaginary part ð Þ ð Þ (76c) þ ebs 2 k 2�2 ð Þ þ � � �� e k� k�� 2� k�� f 0 1 � 2 displaying the imaginary part Im" 2 2 coth csch ¼ 2� � � 0 k� �0 � �0 �0 þ k 1 � � k X¼ � ��0 1 1 k�� (71) Im"s ð� Þ csch 4� 2 2 k� � ¼ � � � 0 k 1 � 0 � þ X¼ which is again a reflection the instability of strong electric 2� k�� k� fields, as confirmed in Fig. 7 by its suppression in domi- 1 coth e� (77) � � � � nantly magnetic fields. The polarization function in general � 0 0 � field configurations can also be exhibited in the quasistat- which is positive only for dominantly electric (0

065026-11 67 LANCE LABUN AND JOHANN RAFELSKI PHYSICAL REVIEW D 81, 065026 (2010)

FIG. 6 (color online). The dielectric function for spin-1=2 (left panel) and spin-0 (right panel) quantum fluctuations. The value is the correction to the Maxwell 1, and as in BI theory, a dominantly magnetic field (below the dotted line) gives a negative correction suppressing the Maxwell energy-momentum. ground are obtained immediately from Eqs. (73) and (74) Numerical evaluations of the EH corrections to the by multiplication with 1=2 and insertion of an alternating spinor and scalar dielectric functions for general field 1 k in the sum. The� weak field expansion is strengths are displayed in Fig. 6.�" again reflects the ð� Þ unusually square character of the EH integrals, though in agreement with Born-Infeld theory, dominantly magnetic � e2 fields suppress (�" <0) the Maxwell tensor. However, the �"s 6 14S � 1440� m4 boundary for which this magnetic suppression is present differs between fermionic and scalar electrodynamics, � e4 S2 P 2 being approximatelyS P 2 in the former case andS 8 279 77 ... (78) þ 20 160� m ð þ Þ þ P in the latter. Indeed,/ the transition to augmentation/

FIG. 7 (color online). The imaginary part of the dielectric function for spin-1=2 (left panel) and spin-0 (right panel) quantum fluctuations. For spin-1=2, the dielectric function is consistently positive; however, as indicated on the plot itself in the scalar dielectric function, the imaginary part is positive to the left and negative to the right of the dotted line atS P . The vanishing of�" s for the anti-self-dualS P field configuration is a striking difference from the spinor case, which is suppressed¼� whenS>P 2. Details are ¼� 20 suppressed when�" < 10 � .

065026-12 68 DARK ENERGY SIMULACRUM IN NONLINEAR... PHYSICAL REVIEW D 81, 065026 (2010)

(�" >0) appears to arise in conjunction with the growing ueff 2 2 6 1 8� �Z eBsurf Rsurf 1:48M m � imaginary part, as seen in Fig. 7. ð Þ 2 6 � ugrav ¼ 135 26 9mu� m r r 0 � � e � � � (81) V. KINEMATICAL EFFECTS OFT converting Newton’s constant into the convenient units With numerics providing the magnitude of the induced G 1:48 km=solarmass. The� - cutoff in Eq. (80) cancels vacuum fluctuation, we can accurately evaluate physical against¼ particle massm, making Eq. ( 81) independent of situations in which the modification of the Maxwell both the mass of the particlem and the cutoff� - . energy-momentum tensor has observable consequences. Remarkably, at the surface of a1:5M , 14 km radius star � A fundamental (i.e. vacuum) nonlinearity of electromag- with critical surface fieldB surf B c, the nonlinear- netism is phenomenologically identical to a ponderable electromagnetic effective potential is¼ 34 times the gravita- medium with nonlinear dielectric response. As may be tional potential, resulting in a large repulsive, quasi- verified by direct calculation (see also discussion in Lorentz-scalar potential for charged particles entering the Sec. 8 of [17]), the formal structure of the Lorentz force strong field region. is therefore unaltered For a relativistic particle, a consistent treatment and thorough discussion of the force due to vacuum fluctua- f� j F��; (79) tions in a strong magnetic field are given in [9]. ¼ � as it is dictated by the necessity of gauge invariance in the VI. DISCUSSION AND CONCLUSIONS coupling of EM potentials to charged matter. This point is The central motivation of this study is the observation to be contrasted with the modification of particle that externally applied fields in nonlinear electromagne- properties. tism have a dark energy-like contribution to the energy- Violation of the superposition principle [23,24] entails momentum tensor. We therefore examined the physics an interaction between the background field and the field giving rise to the trace of the energy-momentum tensor generated by the charged matter. Such an interaction can be as an avenue of insight into the origin of the observed dark extracted by careful study of the Lorentz force [9], but it is � energy in the universe. AsT � in our study is generated by more easily evaluated using the weak field expansion of the quantum-induced nonlinearity of the electromagnetic field EH effective potential. - the physics of dark energy is accessible to laboratory As an example, take a large (r > �e) charged sphere in a experiment probing electromagnetism at high fields. strong background magnetic field, which could provide a We derived the energy-momentum tensor for general rough model for an� particle in the atmosphere of a highly nonlinear electromagnetic theories and emphasized the magnetized neutron star. In the rest frame of a nonrelativ- form Eq. (10). We considered the relationship of the trace istic charged probe particle, we take the background mag- with the matter condensate and obtained a result, Eq. (36) ~ netic field as constant B B^z and the electric field as the which amounts to removal of the leading term inS in ¼~ 2 particle’s Coulomb field E Ze r=r^ . Integrating the en- c� c . This reduces the numerical results by a factor of ¼ 00 ergy of the combined field configuration,T , over the habouti 100 and along with this, the physical dark energy volume with a short distance cutoff at the Compton wave- effect of the energy-momentum trace is greatly reduced. length�- , the leading contribution is the traceT u : ! eff Employing the resummation technique introduced in [14], we numerically evaluated the deviations from the 2�2 2�2 4� Maxwell tensor, the dielectric function and the trace for 4 P 2 S 2 2 ueff d x 4 7 4 4 - ZeB ; Born-Infeld electromagnetism and the Euler-Heisenberg ¼ 45me ð þ Þ ¼ 45me 3� ð Þ Z effective action with both Fermi and Bose matter fields. (80) We believe that these are the first presentations in literature of the matter condensate and energy-momentum trace keeping only the nonlinear-sourced cross terms. The coef- arising from the Euler-Heisenberg action at field strengths @Veff ficient of the Maxwell energy S also induces cross well beyond critical. The dielectric function is found in � @ terms subleading atO � 3 . The cutoff arises since at both BI and EH to suppress the Maxwell tensor in the ð Þ distances shorter than�- we must use quantum dynamics presence of dominantly magnetic fields. The dielectric to describe the probe particle, consideration of which response of the vacuum thereby enhances the observable would be inconsistent with the classical particle dynamics. consequences of the presence of the energy-momentum This interaction energy is positive, independent of the trace. sign of the charge, and comparable to the gravitational The Born-Infeld theory is at first sight an interesting potential of a neutron star with dipolar magnetic field. As source of energy-momentum trace. However, a lower limit 1 u is negative and r � and the effective (scalar) po- on the limiting BI electric field strength obtained 30 years grav / tential goes withB 2 r 6, ago from the study of precision atomic and muonic spectra / �

065026-13 69 LANCE LABUN AND JOHANN RAFELSKI PHYSICAL REVIEW D 81, 065026 (2010) [23] requiresM 2=e 1:710 22 V=m, implyingM However, the lesson for cosmology is that the external 60 MeV. Contemporary�g 2 experimental results, if� an- field framework reveals a physical interpretation of the alyzed with the objective� to set a limit on the BI scale, energy-momentum trace as the observable—and hence would very probably push this limit further up. Asg 2 of gravitating—energy of a false vacuum. As the external the electron in strong fields is in itself a project requiring� field is global in extent, this vacuum energy is indistin- study of the two-loop Lagrangian [25], such analysis takes guishable from a cosmological constant. Thus, the QEDT us far beyond the scope of this paper, and we leave the provides an experimentally tangible simulacrum of the investigation to the future. The scale of the energy- ‘‘cosmological constant’’ momentum trace of BI-type must be rather large and the We left open in this work the question of how the effect at best comparable to the effect of vacuum fluctua- structure of the QCD vacuum, which is very strongly tions expressed by the Euler-Heisenberg effective action. deformed by glue and quark fluctuations, relates to the Even though the energy-momentum trace is suppressed by traceT and responds to an applied electromagnetic field. 2 6 the QED coupling, �=� 6 10 � (see Sec. IV), the Some discussion of this question, including its relation to effective scale 40mð 20Þ MeV¼ � implies that quantum fluc- dark energy, has been already offered [4]. Combining the e ’ tuations remain dominant compared with any other current quantum vacuum with general relativity is a very delicate theoretical framework, given the experimental constraints. question [3]. Our report establishes an important connec- This is consistent with BI being a high-cutoff theory tion, tying already-recognized quantum vacuum effects to arising from more fundamental matter properties. Hence, dark energy. we devoted the largest part of this report to expanding our To summarize, we have studied the energy-momentum prior study of electron fluctuations in the vacuum, the tensor of nonlinear electrodynamics emphasizing an ex- energy-momentum trace has previously been discussed as plicit relationship of the dark energylike trace of the a signal of vacuum deformation [9]. The energy density in energy-momentum tensor Eq. (10b) to the nonlinearity of the trace is then interpreted as the shift of the vacuum the theory. In the consideration of electrodynamics as a energy induced by the applied field, and in concordance quantum gauge theory, the connection provided a new with this interpretation, the trace is positive definite when derivation of a nonperturbative identity between the extracted correctly from the effective action. The smooth energy-momentum trace and the gauge and matter con- BIT highlights the extraordinary form of the quantum- densates, Eq. (36). The Euler-Heisenberg effective action induced trace. Prior considerations of the analytic structure provided a natural example for the numerical evaluation of of the Euler-Heisenberg effective action had not made the condensate and energy-momentum trace, the results of apparent the near singular boundaries present in the con- which are displayed for both fermionic and scalar fields in densate and the trace. The present evaluations suggest Figs. 3 and 5. Finally, we briefly explored the implications further investigations into the strong field vacuum phase of an energy-momentum trace for charged-particle structure. kinematics. We provided further the first complete numerical calculations of the electron-positron condensates of spinor and ACKNOWLEDGMENTS scalar QED for arbitrary fields and have addressed the contradictory statements in literature relating to claims We thank Professor D. Habs, Director of the Cluster of thatT and m c� c are equal, and we found in our Excellence in Laser Physics—Munich-Center for nonperturbative� h studyi a clear difference originating in the Advanced Photonics (MAP) for hospitality in Garching scale dependence of charge renormalization. We empha- where this research was in part carried out. This work size that our evaluation of the condensate and of the was supported by the DFG Cluster of Excellence MAP energy-momentum trace are completely independent of (Munich Centre of Advanced Photonics), and by a grant renormalization procedure. from the U.S. Department of Energy DE-FG02- The energy-momentum trace of QED and its effects 04ER41318. exist anywhere and everywhere an electromagnetic field is present. For instance, the observed dark energy density APPENDIX: IMPROVING CONVERGENCE OF EULER-HEISENBERG INTEGRALS � 4 10 3 2:325 meV 6:09 10 � J=m (82) In this appendix, we display the steps in the transforma- 4�G ’ ð Þ ’ � tion of the proper time integrals Eqs. (25) and (26) into the more rapidly convergent representations used for numerics requires a magnetic field of 108 T. To claim such a mag- in the text. netic field spanning most of the universe has gone as yet The effective action for electrodynamics displays non- undetected would be extremely far-fetched, and the mea- analyticities that, generating an imaginary part of the ac- sured intergalactic magnetic field on the order of�G is tion, are associated with the instability of the vacuum. clearly insufficient to explain the observed dark energy. However, our method of resumming the poles is very

065026-14 70 DARK ENERGY SIMULACRUM IN NONLINEAR... PHYSICAL REVIEW D 81, 065026 (2010) useful for improving the overall convergence of integrals 2 x 4 1 1 1 of the Euler-Heisenberg form, and we start with the case of x cotx 1 2x 2 2 2 2 2 ; � ¼� 3 þ k 1 k � x k � only a magnetic ﬁeld being present, for which X¼ � we obtain Eq. (44). m2 ds � 1 For general ﬁelds, the proper time integrals are rewritten m c c 2 2 eBs cotheBs 1 (A1) � h i ¼ 4� 0 s ð � Þ Z using the Sitaramachandrarao identity [Eq. (6) in [26]] is analytic on the real axis. We use the (subtracted) mer- 2 2 x y 3 1 1 coth k�y=x omorphic expansions, xy cothx coty 1 � 2x y 2 ð 2 2 Þ ¼ þ 3 � k 1 k� x k � X¼ þ 1 1 2 3 1 1 coth k�x=y x cothx 1 2x 2 2 2 (A2) 2y x ð Þ (A8) � ¼ k 1 x k � k� 2 2 2 þ þ k 1 y k � X¼ X¼ �

2 with the result x 1 1 1 2x 4 : (A3) 2 2 2 2 m4 ¼ 3 � k 1 k� x k � T QED ð Þ þ ð Þ Ia I b ; (A9a) X¼ ¼ 2�2 ð þ Þ Inserting Eq. (A2) in Eq. (A1) we obtain 4 3 1 2 s 1 coth k�b=a Ia e a b dss e� 2 ð2 Þ 2 ; (A9b) 2 2 m2s ¼ 0 k� k � eas m eB 1 e k 1 m c� c ð Þ 1 ds � : (A4) Z X¼ ð �ð Þ Þ 2 2 2 1 coth k�a=b � h i ¼ 2� 0 k 1 eBs k� 4 3 1 2 s Z X¼ ð Þ þ ð Þ Ib e b a dss e� 2 ð2 Þ 2 : (A9c) ¼ 0 k 1k� k � ebs All terms are individually absolutely convergent, so we Z X¼ ð þð Þ Þ reorder the sum and integral following the procedure in Rescaling converts these expressions to those found in [14]. After rescalings sk�=eB, thek sum is evaluated Eq. (46). ! in closed form and we obtain The quasistatistical representation ofI a in Eq. (47) is derived by exchanging the sum and the integral in order to m4 ln 1 e �0s � 1 � integrate by parts, m c c 2 ð � 2 Þ ds; (A5) � h i ¼ � 2� �0 0 1 s Z þ 1 1 1 x i� Ia b ds s ln � þ which is Eq. (41). ¼ � 0 � þ 2 �1 x i� �� Equation (A3) allows us to remove the quadratic term in Z þ � 1 k�a 2 Eq. (40), and, rescaling and resumming, we find Eq. (42). coth e k� = ea m s: (A10) b �ðð Þ ð ÞÞ � k 1 � � A further integration by parts results in X¼ m4 s arctans We break up the coth function: T f 1 � ds (A6) 2 �0s �x ¼ 2� 0 e 1 e x Z � cothx e� : (A11) ¼ 1 e 2x which is thea 0 limit of Eq. ( 47), but we retain the form � 1 � � ! X¼� Eq. (42) for clarity in the associated discussion of signs. Expanding the denominator as a power series, the sum in Turning now toB 0, i.e. electric field only witha Eq. (A10) becomes E , we see in the meromorphic¼ expansion ! j j k� exp m2s 2n 1 � eb : (A12) 2 1 1 x cotx 1 2x k;n;� � ea ð þð þ � Þ Þ 2 2 2 X � � � ¼ k 1 x k � X¼ � Since exp k�b=a <1, then sum is absolutely conver- the singularities that indicate the instability of the system gent, thoughð� slowlyÞ when b=a 1. We can exchange the to produce real pairs. We assign to the mass a small order of summation and do the�k sum: imaginary componentm 2 m 2 i� so that ! þ �H � 1 1 1 x i� e� 2 2 m2s Ia b ds s ln � þ m eE 1 e� �H m c� c 1 ds ; ¼ � 0 � þ 2 �1 x i� ��n;� 1 e � � ð 2 Þ 2 2 Z þ � X � � h i ¼ 2� 0 k 1 eEs k� i� Z X¼ ð Þ � ð Þ þ (A13) (A7) 2 2 with� �m =ea andm H � the inner expression in whence resummation produces Eq. (43). Removing the Eq. (A12� ), thus obtaining Eq. (47). leading term in meromorphic expansion for the case of For the scalar case, Eq. (26), we require identities paral- the electric field by use of leling those used in the spinor integrations:

065026-15 71 LANCE LABUN AND JOHANN RAFELSKI PHYSICAL REVIEW D 81, 065026 (2010)

k 2 2 k 1 2 4 1 1 1 x y 3 1 1 xcschx 1 x 2x 2 2ð� Þ2 2 (A14a) xycschx cscy 1 � 2x y ð� Þ � ¼� 6 � k 1 k� x k � ¼ þ 6 � k 1 k� X¼ ð Þ þ X¼ k 2 1 1 csch k�y=x 2x 2ð� Þ2 2 (A14b) 2 ð 2 2 Þ ¼ k 1 x k � � x k � X¼ þ þ k 3 1 1 csch k�x=y 2xy ð� Þ 2 ð 2 2 Þ : (A16) þ k 1 k� y k � k ¼ � 1 2 4 1 1 1 X x cscx 1 x 2x 2 2ð� Þ2 2 (A15a) For the further resummation resulting in the statistical � ¼ 6 þ k 1 k� x k � X¼ ð Þ � representation, the csch function is expanded analogously k 2 1 1 2x ð� Þ : (A15b) 2 1 2 2 2 cschx e x e x e 2nx (A17) ¼ k 1 x k � 2x � � � X¼ � ¼ 1 e � ¼ n 0 � X¼ The representation for general ﬁelds uses an identity but the absence of cosh in the numerator means no sum closely related to Eq. (A8) [see Eq. (20) of [26] ]: over� is introduced. The remaining procedure is the same.

[1] E. Komatsu et al. (WMAP Collaboration), Astrophys. J. 181 (1977). Suppl. Ser. 180, 330 (2009). [15] W. Dittrich and H. Gies, Phys. Rev. D 58, 025004 (1998). [2] P. Serra, A. Cooray, D. E. Holz, A. Melchiorri, S. Pandolﬁ, [16] G. M. Shore, Nucl. Phys. B460, 379 (1996). and D. Sarkar, Phys. Rev. D 80, 121302 (2009). [17] I. Bialynicki-Birula and Z. Bialynicka-Birula, Quantum [3] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989). Electrodynamics, (Pergamon, Oxford, 1975). [4] R. Schutzhold, Phys. Rev. Lett. 89, 081302 (2002). [18] J. Rafelski, L. P. Fulcher, and W. Greiner, Nuovo Cimento [5] H. Euler and B. Kockel, Naturwissenschaften 23, 246 Soc. Ital. Fis. B7, 137 (1972). (1935); H. Euler, Ann. Phys. (Berlin) 26, 398 (1936); W. [19] J. B. Kogut and D. K. Sinclair, Phys. Rev. D 73, 114508 Heisenberg and H. Euler, Z. Phys. 98, 714 (1936); For (2006). translation see arXiv:physics/0605038. [20] I. A. Shushpanov and A. V. Smilga, Phys. Lett. B 402, 351 [6] J. S. Schwinger, Phys. Rev. 82, 664 (1951). (1997). [7] M. Born and L. Infeld, Proc. R. Soc. A 144, 425 (1934). [21] L. Labun and J. Rafelski, Phys. Rev. D 79, 057901 (2009). [8] E. S. Fradkin and A. A. Tseytlin, Phys. Lett. 163B, 123 [22] G. V. Dunne, H. Gies, and C. Schubert, J. High Energy (1985); A. A. Tseytlin, Nucl. Phys. B501, 41 (1997). Phys. 11 (2002) 032. [9] L. Labun and J. Rafelski, Phys. Lett. B (in press). [23] J. Rafelski, L. P. Fulcher, and W. Greiner, Phys. Rev. Lett. [10] M. S. Chanowitz and J. R. Ellis, Phys. Rev. D7, 2490 27, 958 (1971); G. Soff, J. Rafelski, and W. Greiner, Phys. (1973); Phys. Lett. 40B, 397 (1972). Rev. A7, 903 (1973); J. Rafelski, W. Greiner, and L. P. [11] R. J. Crewther, Phys. Rev. D3, 3152 (1971);4, 3814(E) Fulcher, Nuovo Cimento Soc. Ital. Fis. B 13, 135 (1973). (1971); Phys. Rev. Lett. 28, 1421 (1972). [24] C. A. Dominguez, H. Falomir, M. Ipinza, M. Loewe, and [12] S. L. Adler, J. C. Collins, and A. Duncan, Phys. Rev. D 15, J. C. Rojas, Mod. Phys. Lett. A 24, 1857 (2009). 1712 (1977). [25] V.I. Ritus. in Proc. Lebedev Phys. Inst., Issues in Intense- [13] J. C. Collins, A. Duncan, and S. D. Joglekar, Phys. Rev. D ﬁeld Quantum Electrodynamics,Vol. 168, edited by V.I. 16, 438 (1977). Ginzburg (Nova Science Pub., NY, 1987). [14] B. Mu¨ller, W. Greiner, and J. Rafelski, Phys. Lett. 63A, [26] Y.M. Cho and D. G. Pak, Phys. Rev. Lett. 86, 1947 (2001).

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APPENDIX D

QED Energy-momentum Trace as a Force in Astrophysics

QED Energy-momentum Trace as a Force in Astrophysics Labun, L. and Rafelski, J. Physics Letters B687 (2010) 133-138 doi: 10.1016/j.physletb.2010.02.083 arXiv:0810.1323 [hep-ph]

Copyright (2010) by the Elsevier Reprint permission granted under License Agreement 2785561090185 Eﬀective 10 November, 2011 73

Physics Letters B 687 (2010) 133–138

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

QED energy–momentum trace as a force in astrophysics

a,b a,b, Lance Labun , Johann Rafelski ∗

a Department of Physics, University of Arizona, Tucson, AZ 85721, USA b Department für Physik der Ludwig-Maximilians-Universität München, Germany

article info a b s t r a c t

Article history: We study the properties of the trace T of the QED energy–momentum tensor in the presence of quasi- Received 22 July 2009 constant external electromagnetic fields. We exhibit the origin of T in the quantum nonlinearity of the Received in revised form 17 February 2010 electromagnetic theory. We obtain the quantum vacuum fluctuation-induced interaction of a particle with Accepted 26 February 2010 the field of a strongly magnetized compact stellar object. Available online 11 March 2010  2010 Elsevier B.V. All rights reserved. Editor: W. Haxton

Keywords: Supercritical stellar magnetic ﬁeld Casimir modiﬁed Lorentz force Conformal symmetry breaking Euler–Heisenberg nonlinear electrodynamics

1. Introduction making strong-field QED effects possible in the laboratory [11]. Magnetic fields of strengthB � Bc are also encountered in Quantum electrodynamics (QED) in (quasi-)constant, homoge- the study of supernovae and post-main sequence stellar objects neous external electromagnetic (EM) fields provides an opportu- [12–15], and we show here how the vacuum energy leads to a nity to study the properties of the vacuum state structure un- novel often repulsive force between the stellar magnetic field and der the conditions of extreme external fields. In the presence of a charged particle, influencing matter accretion and the supernova an electromagnetic field that varies negligibly on the space–time bounce. Much of what we present here is a general property of any scale of the electron–positron fluctuations in the vacuum λ-c = nonlinear theory of electrodynamics [16], applicable also to Born– h/mec leads to an effective nonlinear electromagnetic theory via the¯ Euler–Heisenberg (EH) effective action [1–10]. The physical Infeld electromagnetism, for example. We achieve considerable observables and effective action induced by quasi-constant exter- simplification and insight exploiting a common feature of any non- nal electromagnetic fields are well-defined, because QED is an linear electromagnetism, namely the presence of a dimensioned 2 field scaleE c m /e which we express using a mass scalem. We infrared-stable theory in which the electron massm e is the key = scale parameter. write the (nonlinear) effective electromagnetic action The requirement of Lorentz symmetry admits only one essentially new contribution in the final expression for the energy– µν S P momentum tensorT , a vacuum energy term proportional 4 µν V eff S m feff , , (1) tog . This term is similar to Einstein’s form of dark energy, the ≡ − + �m4 m4 � cosmological constant Λ. The related repulsive anti-gravity like effect of the energy of vacuum fluctuations may become accessible 1 αβ κλ 1 2 2 S gκα gλβ F F B E , (2) to laboratory experiments: pulsed laser technology is advancing := 4 = 2 − rapidly towards the ‘critical’ field strength � � 1 2 18 9 P g g F αβ F κλ E B (3) Ec Bc m /e 1.3 10 V/m 4.4 10 T, κα λβ = = = × = × := 4 = · � as a function of the (Lorentz) scalar S and pseudo scalar P. In * Corresponding author at: Department of Physics, University of Arizona, Tucson, S AZ 85721, USA. Eq. (1) feff contains solely contributions nonlinear in , exclud- E-mail addresses: [email protected] (L. Labun), ing a possible linear term, e.g. S lnm /µ where µ is another scale. [email protected] (J. Rafelski). When such term is included we omit the bar onf eff.

134 L. Labun, J. Rafelski / Physics Letters B 687 (2010) 133–138

2. Energy–momentum trace purposes, the form of Eq. (9) is confirmed by the observed stability of the QED vacuum and work claiming otherwise will need to We study the energy–momentum tensor for an effective non- address that important issue. linear action (g detg µν ) In fact, the relative sign in Eq. (9) agrees with Eq. (5) of Ref. [19] = and Eqs. (35) in [22] with the recognition that the (fermion) Gell- µν 2 δ 4 Man–Low β-function in QED is positive definite. Our equation (9) T − d x √ gV eff (4) = √ g δgµν � − (and the more explicit form of T , Eq. (11) below), agrees with [20], − which result is a bit surprising since it follows from the clearly µν ∂ V eff ∂ V eff εT gµν V S P . (5) contradictory supposition that T m∂ V /∂m. Moreover, there Max eff S P eff = − � − ∂ − ∂ � are quite a few other instances in= literature where the first term in Eq. (9) is omitted. The dimensionless dielectric function ε ∂ V eff/∂S 1 in the µν S = − → classical MaxwellT Max limitV eff . The form of Eq. (5) agrees with Eq. (A3) in [17] and Eq. (4.17)→ in− [18]. 3. Strong fields simulacrum of dark energy The first term in Eq. (5) is traceless. The second term in Eq. (5) provides as noted the only possible covariant extension and is Applying Eq. (9) with the Euler–Heisenberg–Schwinger effec- identically the trace of the energy–momentum tensor. Using Eq. (1) tive action (using Schwinger’s notation and units in which α e2/4π [2]) gives an explicit formula for the trace, = µ ∂ V eff ∂ V eff dV eff Tµ T 4 V eff S P m . (6) 2 ∞ S P 2α m 2 coth(ebs) 1 ≡ = − � − ∂ − ∂ � = − dm T S ds e m s e2ab (11) 2 − 2 T = 3π − 4π � � tan(eas) − s � Separation of the trace from the off-diagonal Maxwell-like part 0 of the energy–momentum tensor isolates the field induced, gravi- wherein the invariant magnetic- and electric-like field strengths tating energy–momentum of the vacuum. are Before exploring the physical consequences, we must pause and clarify the precise meaning of T . Terms linear in the invariant S b2 S2 P2 S B2, a2 S2 P2 S E2, do not contribute to the right side of Eq. (6) since they cancel = � + + → = � + − → reducing as indicated to the classical magnetic and electric fields explicitly in the middle parentheses. Only nonlinear (in S, P) EM- when one invariant vanishes. theories can have an energy–momentum trace and the bar above In numerical evaluation of the energy–momentum tensor for V reminds us of this. This is another way to say that QED with eff arbitrarily strong fields we employ the method developed in [6]. massive electrons is ab initio not conformally symmetric. Massive Consider first the stable field configurationB 0,E 0. The sub- QED does not share the more challenging issues, such as confor- tracted meromorphic (i.e. residue) expansions�= of the= function mal symmetry breaking, surrounding parallel efforts in quantum chromodynamics (QCD) [19]. ∞ 2x2 x2 ∞ 1 2x4 A better understanding of these remarks is achieved in QED by x cothx 1 (12) − = x2 k2π 2 = 3 − (kπ)2 x2 k2π 2 connecting T to the Dirac (electron–positron) condensate induced �k 1 �k 1 = + = + in the vacuum [20], which is directly related to the effective action display the stabilizing change in sign following the second subtraction. The sums and integrals are absolutely convergent, so we may 0 dV eff m ψψ im tr S F S m . (7) resum the resulting series, obtaining − � ¯ �= − F = dm � � The middle expression of Eq. (7) exhibits the condensate as the m4 ∞ V (B) dz ln z2 1 ln 1 e β�z , (13) difference between normal ordering of operators in the no-field eff 2 − = 16π β� � + − (also called perturbative) vacuum and the with-field vacuum. The 0 � � � � argument of the derivative in Eq. (7) and in Eq. (6) is not exactly 4 ∞ β z the same: The difference in format between Eqs. (6) and (7) is due m ln(1 e− � ) m ψψ − dz > 0, (14) to the leading linear term in S ¯ B 2 2 − � � = − 2π β� � 1 z 0 + dV eff dV eff 2α m m S. (8) 4 ∞ 2 β z dm dm 3 m z ln(1 e− � ) = + π T dz > 0, (15) B 2 − 2 Combining Eqs. (6), (7), and (8), we obtain the well-known re- = − 2π β� � 1 z 0 + lation [21] 2 in which β� πm /eB π/(B/Ec). Eq. (13) presents the (renor- 2α = = T S m ψψ (9) malized) effective action also seen in Ref. [6]. Numerical evalu- = 3π + � ¯ � ations of the condensate (14) and trace (15) in the presence of 2 magnetic and electrical field (real part only) are shown in Fig. 1. 2α 2 2 3 7P 4S O α . (10) We discuss elsewhere [16] the case that bothE,B are non-zero, = 45m4 + + e � � � � how real and imaginary parts contribute together for electric fields, Eq. (9) displays two contributions to the trace of the energy– and the case of spin-0 matter fields. For fields way above critical momentum tensor: gauge field and matter field fluctuations. There the results presented in Fig. 1 are not accessible in practice by di- is an important exact cancellation between these two terms. Were rect integration of the proper time representation (11). the first term in Eq. (9) (erroneously) omitted, the trace T would For the electric field case, the corresponding meromorphic ex- be that much greater, and for any applied magnetic field the pansions ofx cotx show poles on the reals-axis. We assign to the 00-energy-density-component of the energy–momentum tensor mass a small imaginary componentm 2 m2 i�, replacing in would be negative. The QED vacuum would be unstable, and the Eqs. (14) and (15) the denominatorz 2 →1 +z2 1 i�. Thus, naive perturbative QED vacuum could reduce its energy by spon- in the presence of an electric field, there+ is→ also nonperturbative− + taneously generating a state with magnetic field. For all practical imaginary contribution to T . 75

L. Labun, J. Rafelski / Physics Letters B 687 (2010) 133–138 135

ergy rivals cosmological dark energy λ atB d 108 T, the scale of the largest static laboratory fields. = Energy–momentum sources in Eq. (16) involving local dark energy-like contributions have only recently been studied [24–27]. To see that the effects of local T are anti-gravitational just like cosmological λ, we inspect the Oppenheimer–Volkov equation dp G M 4πr3 T i T 0 T i + i (17) dr = − c2 0 + i r(r 2GM) � � − whereM (r) r T 04πr 2 dr andT i p λ is assumed isotropic. = 0 � � i = − λ does not contribute� to the first term which is always non- 0 i negative:T 0 Ti ( ρ λ) ( p λ), but λ can make the second + 3 i= + + − µ term (M 4πr Ti ) change sign. Contributions toT ν proportional µ + tog ν (like λ) thus weaken the pressure gradient and support heavier stars than otherwise expected. We have checked using Eq. (17) that direct gravitational modifications to the mass-radius relation for compact stellar objects T 4 remain negligible, as might be expected. At 60B c , 0.8me 1.14 1025 erg/cm3, 8 orders of magnitude smaller than= the pres-= sures× expected in the high density nuclear matter in the core of a post-main sequence star [15]. However, this energy density is 2–4 orders of magnitude larger than the gravitational potential energy density of infalling stellar plasma. Thus while the insignificance of the QED T in gravity is a consequence of its objectively small energy density, the anti-gravitational effect of T suggests a closer study of the force experienced by individual particles in nonlinear Fig. 1. The condensate m ψψ (top) and the trace of energy–momentum tensor µ − �4 ¯ � electromagnetism is necessary. Tµ (bottom) in units ofm , as a function of magneticB (red) and electricE (blue, dashed and solid) field strengths. The negative of the electric field result is plotted where appropriate. The dotted (forE) and dashed-dotted (forB) lines show the 4. Particles in overcritical quasi-constant fields weak-field expansions up toE 6,B 6. (Color online.) Forces present in the dynamical case can be much greater than From now on in this work we address strong magnetic fields. those observed when the interacting bodies are studied in the hy- T in the presence of a magnetic field is positive for any given drostatic equilibrium of Eq. (17). The electromagnetic force deter- fieldB, in contrast to the negative of the condensatem ψψ . The mining individual particle dynamics does not make relevant contri- ¯ manifest signs of the two expressions (14) and (15), which� � deter- butions, and the vacuum fluctuation-induced force contributes even mine the physics outcome of this investigation justify the time and less to Eq. (17). Though very much smaller than Maxwell’s force effort spent showing how T does not include the term linear in S, this force can be stronger than gravity and at times more relevant whilem ψψ does. Clarification of this exclusion is necessary since than the linear order force of Maxwellian electromagnetism. We ¯ as noted� T and� m ψψ are often conflated in literature. will describe its features relevant to the charged particle dynamics ¯ The trace T gravitates,� � just as the Casimir energy does [23]. Be- within collapsing stellar objects. cause in the Euler–Heisenberg–Schwinger calculation the ‘constant’ Consider that the total (‘t’) electromagnetic energy–momentum µν external field is global in extent, this energy–momentum is mani- tensorT due to both an external field (‘e’) and a probe charge t µν fested in the form of a cosmological constant. In contrast to matter, (‘p’) includes also an interaction energy–momentumT int , for which the particle pressure acts outwards, the pressure part of µν µν µν µν T T T T (18) energy–momentum tensor described by T /4 acts inwards. This t = e + p + int is a general feature of any ‘false’ vacuum− state: the outside true µν µν with tensorsT e ,T p defined by the forms they take in isolation vacuum the squeezes the false, higher energy density vacuum out from each other. When the external field is much larger than the of existence. field of the probe particle, the electromagnetic energy–momentum T The sign reversal of pressure (compared to pressure of regu- tensor is expanded in the displacement tensorK µν lar matter) overwhelms the gravity of the positive energy density, providing the anti-gravity effect associated with dark energy. The µν ∂ V eff µν ∂ feff T K F , (19) similarity of to the cosmological constant was noted before by = −∂ Fµν = − ∂ Fµν Schützhold [19] and can be made explicit in the Einstein equation around the dominant contribution of the external field, by separating the trace, Tµν Tµν gµνT /4 = − µν αβ µν µν ∂ T Kp � T T (20) 1 1 T λ t e αβ Rµν gµν R Tµν gµν . (16) = + ∂ K �e 2 + · · · 8π G � − 2 � = − − � 4 + 2 � � with the subscript ‘e’� reminding that derivatives are to be evalu- � � ated at the external field. With a sign like that of dark energy Λ/4π G λ ( 2.3 meV)4 The energy–momentum tensorT µν is expressed in Eq. (5) in 4.1 10 34m4, T is the dominant contribution≡ � in a domain − e terms of the field tensorF µν , but only the displacement fields of =of space× with strong fields and is naively expected to generate a the probe particle are known explicitly by solving Maxwell’s equa- pressure that sweeps out matter, in analogy with the cosmological tions with source acting constant at large scales and pushing the universe apart. For µν ν comparison, we note that the magnetic field-induced vacuum en- ∂µ K j . (21) p = p 76

136 L. Labun, J. Rafelski / Physics Letters B 687 (2010) 133–138

By inverting Eq. (19) we obtain Be changes on macroscopic scale though, and the parameter char- acterizing smallness of the effect is 1/mL 10 16 whenB varies αβ 2 − e ∂ F α β α β ∂ feff on the scaleL 4 km. For comparison, the� smallness parameter µν δµδν δ ν δµ µν � ∂ K = − + ∂ F ∂ Fαβ of the vacuum fluctuations arising from Euler–Heisenberg action � � 2 2 2 is (Be/Bc) α/45π . Seeing that vacuum fluctuation effects should ∂ feff ∂ feff . (22) dominate the magnetic dipole interaction for a stellar magnetic + ∂ F µν∂ F ∂ F γ δ∂ F + · · · 5 γ δ αβ fieldB e > 10− Bc , we explore this domain further. This rank 4 tensor transforms the functional dependence from the Turning now to the latter two terms of Eq. (23) that represent field tensor to the displacement tensor. We checked the validity of the additional vacuum fluctuation-induced force, we observe that truncation by numerical evaluation of the derivatives of the action, the gradient in the corresponding last term of Eq. (26) generates which shows that (normalized) higher derivatives are suppressed two contributions: the first as the gradient of Eq. (27) and the T S even when the field is supercritical. A separable contribution of the second as the net change of the slowly-varying coefficient ∂ /∂ µν � µν over the domain of particle’s field. The integrals over the particle’s Maxwell self-energyT p TMax,p of the probe particle is indeed field (27) computed in the particle’s rest frame, the spatial com- found at next order ∂2/∂ K 2 and would be subtracted. However, we ponents of the force (26) on a point charge ρ Zeδ(x) in its rest find the effects of the terms from the second derivative are many p � frame are = orders of magnitude smaller than those from the first derivative T and do not discuss them further here. 1 2 ε 2 ∂ δ f � (ε 1) − Ee CeΦe Se, (28) Using Eq. (22) in Eq. (20) gives Ze � � − + 2 ∂S �� − ∇� �e T � µν µν 2 µν 2 ε ∂ e αβ where � T T ( ε 1) T gµν − F K (23) int ep ep S αβ p 2T T = + − + 4 ∂ 2 ∂ε ε 2 ∂ 1 ∂ε ∂ Ce B 2(ε 1) − (29) where c = −� − ∂S + 2 ∂S2 + 2 ∂S ∂S �

µν µκ νλ νλ µκ µν 1 e αβ depends only on the scalar invariant of the external field Se. On Tep Fe K F Kp gκλ g F Kp (24) = − p + e + 2 αβ the right side of Eq. (28) the gradient applied to the invariant � � S e αβ preserves the correct Lorentz transformation property: although is the Maxwellian interaction withF Kp 2(Be Hp Ee Dp). αβ � � � � the potential Φe appears, Eq. (28) is the gauge invariant correction The latter two terms of Eq. (23) remain after= cancellation· − among· to the linear force f e(E v B ) computed in the particle’s the order α terms, and despite being order α3 the last term (1 � �e � � e rest frame. = + × ε)(∂T /∂S) (∂ f /∂S)(∂T /∂S) is kept for now. − eff The component of Eq. (28) proportional toC is qualitatively We view= the net force (density) acting on the charged probe e different because, being proportional the gradient S, it allows particle entering the domain of the external field in the usual way, � the transfer of energy from the magnetic field to in-falling∇ parti- requiring that inertial resistance balance any breach of the conser- cles. This property is in contrast to the first term in Eq. (28) which vation of the field energy–momentum, produces a tiny change in the effective linear force (per mille at µ µν µν µ f ∂ν T jν F δ f (25) Be Bc ). ≡ − int = e + = µν p µν The weak-field expansion ofC e Eq. (29) using that ∂ν Tep jν Fe is the force obtained within Maxwell’s − = 2 linear electromagnetism. Here 2 8α Be 6 2α Ce B � 1 (30) c 45 B 7 45 µ 2 p µν µν 2 π � − � c � � + π + · · ·� + · · ·� δ f �( εe 1) jν Fe Tep ∂ν(εe 1) − − − obtained from the Euler–Heisenberg effective action, is usable up T µ 2 ε ∂ e αβ toB e � 0.1B c . The expansion shows that the predominant con- ∂ − F Kp (26) − 4 ∂S � αβ tribution to the gradient force is T , as the leading constant in �e � Eq. (30) is traced to ∂2T /∂S2. Nonperturbative computation re- with equality only approximate on account of the finite order ex- � quiring employment of the nonperturbative numerical methods pansions inT µν Eq. (20) and α Eq. (22). presented of the coefficients 1, ∂T /∂S, etc., shows that the µν e αβ ε The conventional contributions ofT ep andF αβ Kp , i.e. the force persists in the considered high− magnetic field domain despite first term of Eq. (23), to the force on the particle are obtained what the perturbative expansion suggests. by integrating over a covariant hypersurface, and all frames being equivalent this integration is done most conveniently in the rest 5. Particle dynamics in a stellar magnetic field frame of the particle, allowing us to consider solely theT 00 component As an application, we consider the dipole field of a strongly magnetized star. We are studying the forcef in the rest frame of 3 i j 3 i j ij 3 r d x E Dp d x Φe Φp δ d x Φeρp (27) the particle and in order to allow that the particle has a velocity � e = − � ∇ ∇ = � relative to the star, we need to Lorentz-transform the field in the using that D ( Ze/r2)r (for a spherical charge) has only one star’s rest frameB oriented at an angle ψ from the direction of � p ˆ � component when= choosing spherical coordinates. the particle’s motion. As a consequence of the transformation, the A physical charged particle has also a magnetic moment µ, and field of the star is seen by the particle to have an electric com- � thus a corresponding dipole magnetic field. As is well known this ponent. Specifically,v µ γ (1, 0, 0, β) ( coshy , 0, 0, sinhy ) and part of the force cannot usually compete with the effect of elec- the Lorentz transformed= fields are B =B (cos ψ z γ sin ψ x) and � e � ˆ ˆ trical particle charge. However, in the present context we reach E B γ β sin ψ y. Using this in Eq.=(28) we obtain− the force of �e � ˆ beyond the usual Lorentz force to the effect of vacuum fluctua- the= star’s − field on the moving particle. Note that β,y can be positions and it is necessary to check if it is still justified to neglect the tive or negative. magnetic dipole in the strong magnetic field environment of a col- To compare with the gravitational forcef g, we must also lapsing star. From the magnetic dipole interaction energy µ B Lorentz transform it to the rest frame of the moving particle. Al- p � e a force due to the gradient of the external magnetic field� arises.· though general relativistic corrections to the Newtonian potential 77

L. Labun, J. Rafelski / Physics Letters B 687 (2010) 133–138 137

tum compared to the matter which is attracted and thus angu- lar momentum is imparted to the magnetic source due to the mass asymmetry between positively and negatively charged particles.

In summary, we have evaluated the trace T of the QED energy– momentum tensor and demonstrated that its gradient entails a significant and often repulsive force, which can be large compared to gravity, even while the relative energy density of T remains small. Although the magnitude of the usual magnetic force v B � � e is much larger than the vacuum-fluctuation induced correction× (28), only the latter is relevant in consideration of energy transfer to the particle and escape from the gravitational potential well. The requisite energy exchange with a magnetic field, seen in the gradient S, is a consequence of the nonlinearity of the induced � vacuum fluctuations,∇ absent in classical Maxwellian electromagnetism. The quantitative study we present in Fig. 2 for the ratio (31) indicates the force (26) and Eq. (28), should have an impact on matter accretion and stellar collapse dynamics in as-

Fig. 2. The ratiof r / fgrav Eq. (31) for a transverse (ψ π/2) electron of rapidity trophysical situations where strong magnetic fields in excess of 1 = 4 5 y cosh− γ at the surface of a 1.5M , 12 km radius star with magnetic field given B 10− Bc 10 T are known to exist. While treatment of = � � = on the horizontal axis. Using the expansion Eq. (30) forC e produces the depicted the complete dynamical situation is beyond the scope of this values up toB � 0.1B . (Color online.) c work, it is easy to imagine that this force could help the neutrino based transport phenomena [29] to propel the supernova are significant near the stellar objects where such strong magnetic bounce. fields have been inferred, the Newtonian force is a reasonable first estimate modified only by multiplicative numerical factors of order Acknowledgements unity down to a few times the radius of the future neutron star. We obtain the transformation property of the forcef g considering the geodesic in the Schwarzschild metric (see e.g. Eq. (9.32) This work in part supported by the DFG Cluster of Excel- in [28]): transforming to the rest frame of a relativistic particle di- lence MAP (Munich Centre of Advanced Photonics), we thank lates the proper time, multiplying kinetic and total energies by γ 2. Prof. D. Habs for hospitality. Supported by a grant from the US The energy equation for the geodesic therefore preserves its form Department of Energy, DE-FG02-04ER41318. if the same factor γ 2 is included also in the ‘potential’ terms, thus 2 giving the transformationf g γ fg. The ratio of the radial vac- References uum fluctuation force to the�→ Newtonian gravitational force for a transversely moving (ψ π/2) electron is = [1] H. Euler, B. Kockel, Naturwiss. 23 (1935) 246; 9 2 2 H. Euler, Ann. Phys. V 26 (1936) 398; fr Rsurf eBsurf r W. Heisenberg, H. Euler, Z. Phys. 98 (1936) 714. 3Ce Zeβγ Bsurf , (31) 2 2 [2] J.S. Schwinger, Phys. Rev. 82 (1951) 664. fg = � r � � me � γ GM mp � [3] E. Brezin, C. Itzykson, Phys. Rev. D 2 (1970) 1191. illustrated in Fig. 2. [4] Z. Bialynicka-Birula, I. Bialynicki-Birula, Phys. Rev. D 2 (1970) 2341. The ratiof r/ fg Eq. (31) can be large as shown in Fig. 2 due [5] S.L. Adler, Ann. Phys. 67 (1971) 599. 5 [6] B. Müller, W. Greiner, J. Rafelski, Phys. Lett. A 63 (1977) 181. to the weakness of gravity, particularly forB e 10− Bc . The stel- [7] W. Dittrich, H. Gies, Springer Tracts Mod. Phys. 166 (2000) 1. lar magnetic field contribution to the gravitating� energy density [8] Y.M. Cho, D.G. Pak, Phys. Rev. Lett. 86 (2001) 1947; remains relatively small, yet it affects particle dynamics through W.S. Bae, Y.M. Cho, D.G. Pak, Phys. Rev. D 64 (2001) 017303. the coupling to moving charge, through vacuum fluctuation non- [9] U.D. Jentschura, H. Gies, S.R. Valluri, D.R. Lamm, E.J. Weniger, Can. J. 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[24] P.O. Mazur, E. Mottola, Proc. Natl. Acad. Sci. 101 (2004) 9545. F.S.N. Lobo, A.V.B. Arellano, Class. Quantum Grav. 24 (2007) 1069. [25] G. Chapline, Dark energy stars, in: Pisin Chen et al. (Eds.), Proceedings of 22nd [27] R. Chan, M.F.A. da Silva, J.F. Villas da Rocha, Gen. Relativ. Gravit. 41 (2009) Texas Symposium on Relativistic Astrophysics at Stanford University, Stanford, 1835. California, 13–17 December 2004, 205 pp., Proceedings at: http://www.slac. [28] M.P. Hobson, G.P. Efstathiou, A.N. Lasenby, General Relativity: An Introduction stanford.edu/econf/C041213/, arXiv:astro-ph/0503200. for Physicists, Cambridge Univ. Press, Cambridge, UK, 2006. [26] F.S.N. Lobo, Class. Quantum Grav. 23 (2006) 1525; [29] J.W. Murphy, A. Burrows, Astrophys. J. 688 (2008) 1159. 79

APPENDIX E

Strong Field Physics: Probing Critical Acceleration and Inertia with Laser Pulses and Quark-Gluon Plasma

Strong Field Physics: Probing Critical Acceleration and Inertia with Laser Pulses and Quark-Gluon Plasma Labun, L. and Rafelski, J. Acta Physica Polonica B41 (2010) 2763-2783 arXiv:1010.1970 [hep-ph] 80 POLISHACADEMYOFARTSANDSC IENCES AND JAGELLONIAN UNIVERSITY M.SMOLUCHOWSKIINSTITUTEOFPHYS ICS

ACTA PHYSICA POLONICA B Established in 1920 by the Polish Physical Society

DECEMBER

L CRACOW SCHOOL OF THEORETICAL PHYSICS PARTICLE PHYSICS AT THE DAWN OF THE LHC — DEVELOPMENTS IN PARTICLE PHYSICS FROM A 50 YEAR PERSPECTIVE OF THE CRACOW SCHOOL Zakopane, Poland, June 9–19, 2010

RECOGNIZED BY THE EUROPEAN PHYSICAL SOCIETY

CRACOW 2010

JAGELLONIAN UNIVERSITY 81

Editorial Committee

Michał Praszałowicz (Editor) Wojciech Broniowski, Marek Kutschera Wojciech Słomiński

International Editorial Council

A. Białas (Kraków) (Chairman) L. McLerran (Brookhaven) J. Dąbrowski (Warszawa) A. Morel (Saclay) W. Kittel (Nijmegen) T. Ruijgrok (Utrecht) A.D. Martin (Durham) M. Veltman (Ann Arbor)

Address of the Publisher and the Editor: Institute of Physics of the Jagellonian University Reymonta 4, 30–059 Kraków, Poland e-mail address: [email protected] WWW: http://th-www.if.uj.edu.pl/acta

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Vol. 41 (2010) ACTA PHYSICA POLONICA B No 12

STRONG FIELD PHYSICS: PROBING CRITICAL ACCELERATION AND INERTIA WITH LASER PULSES AND QUARK-GLUON PLASMA∗ Lance Labun, Jan Rafelski

Department of Physics, The University of Arizona, Tucson, 85721 USA

(Received October 25, 2010)

Understanding physics in domains of critical (quantum unstable) ﬁelds requires investigating the classical and quantum particle dynamics at the critical acceleration,˙u 1 [natural units]. This regime of physics remains today experimentally→ practically untested. Particle and laser pulse collision experiments reaching critical acceleration are becoming feasible. Ultra-relativistic heavy ion collisions breach the critical domain but are complicated by the presence of much other physics. The infamous problem of radiation reaction and the challenging environment of quantum vacuum instability arising in the high ﬁeld domain signal the need for a thorough redress of the present theoretical framework.

PACS numbers: 03.50.De, 12.38.Mh, 41.60.–m, 41.75.Jv

1. Introduction

Strong applied “external” fields have long been understood to probe physical properties of the quantum vacuum state. Considerable effort was com- mitted to exploration of vacuum decay by positron production in scattering of largeZ nuclei [1] at sub-atomic scale. Though motivated by a different set of questions, the study of ultra-relativistic heavy nuclei collisions and the associated multi-particle production explores the physics of strong fields in quantum chromodynamics [2]. Slowly-varying(λ m 1) external fields of � − arbitrary strength are employed as a formal tool in theoretical investigations to gain insight into non-perturbative structures of quantum theories of matter interactions [3]. The questions posed when these approaches combine are:

∗ Joint report of individual presentations made by both Authors at the L Cracow School of Theoretical Physics “Particle Physics at the Dawn of the LHC”, Zakopane, Poland, June 9–19, 2010.

(2763) 84 2764 L. Labun, J. Rafelski

What happens when a particle collides with an ultra strong, slowly-varying field [4]? Is our theoretical framework able to cope with this domain of physical parameters [5]? The question is more subtle than it may at first appear, because it probes the foundations of the physical concepts of inertia and acceleration. Ernst Mach noted the intricate connection between inertia and acceleration and that the understanding of acceleration requires introduction of an additional inertial reference frame [6,7]. This widely accepted insight is independent of Mach’s proposal to relate inertia to the matter content of the Universe, which is in disagreement with experiment [6]. With a few important exceptions (e.g. [8]), the necessity of the additional reference frame in the framework for accelerated motion has received little attention in past 100 years of modern physics, perhaps since the magnitude of acceleration we have considered has been “small”. We will show that experiments probing “strong” acceleration are feasible today. Consider the least massive particle readily available to experiment, an electron, entering an extended space-time domain where the electromagnetic field nears the “critical” strength 2 3 mec 18 V 9 Ec = = 1.32 10 = 4.41 10 cT, (1) e� × m × withm e the electron mass. According to the Lorentz force equation duα e = F αβu , (2) dτ − m β this is equivalent to the condition α 3 du mec 29 m 1[m ] = 2.33 10 2 . (3) dτ → ≡ e × s � � � � � � � In figure 1, accelerations� � achieved in physical systems of interest are compared in scale. The acceleration of a proton at the surface of a1.5M , � 12 km radius neutron star is “only”g 1.39 10 11 m/s2, which is shown surf � × in the bottom left panel, and the acceleration at the event horizon of a solar mass black hole lies still farther from the critical scale, see bottom right of figure 1. In contrast, accelerations during electromagnetic and strong interactions reach near to unit magnitude, as seen in left panel. The middle panel shows the wide separation of the Planck mass and its correspond- 3 ing acceleration˙uPl =M Plc /� from both Standard Model interactions and astrophysical scales. The remarkable observation that the Earth-bound experiments marked in figure 1 bear upon strong gravitational physics then follows from the Equivalence Principle, which asserts that the physics is the same as if gravity were the cause. 85 Strong Field Physics: Probing Critical Acceleration and Inertia . . . 2765

20 20 10 10 M Unit acceleration 15 Pl 15 10 10

10 10 10 10

105 105 1 laser-electron collider 1 RHIC -5 E -5 .

u [ GeV ] c 10 SLAC 10 10-10 10-10

10-15 10-15

-20 -20 10 neutron star 1 solar mass BH 10

-6 -3 18 30 36 42 48 54 60 10 10 1 10 10 10 10 10 10 10 Mass [ GeV ]

Fig. 1. Circles mark accelerations presently available, either by laboratory experiment or in the Universe. The s highlight yet unattained accelerations. The × surface gravity of the Earth is too far below the scale of the chart to include.

The challenge remains to understand electromagnetic and strong nuclear forces at their critical scales. For the former, neither the classical nor the related quantum electrodynamic (QED) description can be believed to be complete asE approachesE c—the classical theory because radiation reaction becomes non-negligible [9], and the quantum theory because the electromagnetic field becomes unstable to spontaneous pair creation [10,11]. The looming possibility of performing experiments involving critical fields E/E 1 [12] thus requires a fresh look at charged particle classical and c → quantum dynamics for the case of an applied critical field. We survey here the physics associated with the unit-acceleration regime. We will show how the challenge of describing particle dynamics at this scale will lead to deeper understanding of the building blocks of quantum and classical theories, due particularly to the need to render their descriptions consistent and complete. In the wider view encompassing the present theory of acceleration, namely general relativity, the creation and experimentation with large-scale strong fieldsE E will have ramifications for any unified ∼ c description of particle interactions. We first recall in Section 2 the experimental advances of laser pulse technology. In Subsection 2.2 we discuss a light pulse-electron collider as makes an experiment with macroscopic strong fields possible in the foreseeable future. We then recall in Section 3 the conceptual context of (critical) acceleration and present a thorough discussion of radiation reaction and its connection to the understanding of inertia. We connect in Section 4 to the acceleration reached in high energy elementary and heavy ion collisions. 86 2766 L. Labun, J. Rafelski

In Section 5 we address the appearance of critical acceleration in quantum ﬁeld theories and the related vacuum instability. We brieﬂy compare the physics of accelerated observers and accelerated vacuum as it presents itself at this time in Subsection 5.3.

2. High-intensity lasers 2.1. Technological limits Renewed motivation for addressing strong field physics comes from advances in laser technology. The technique of chirped pulse amplification [12] has made possible laser pulses of higher contrast, shorter length, and relativistic intensities where the field amplitudee A is many times the electron | | rest mass. To surpass the exawatt 1022 W/cm2 record intensities already achieved [13], concerted efforts, such as the Extreme Light Infrastructure with intended sustained laser pulse intensity as high as 1024 W/cm2, are in final planning stages [14]. In field strengths, these intensities respectively correspond to E =2.7 10 15 1016 V , (4) | | × − m or in terms of the dimensionless, normalized amplitude commonly used e A e E a0 | | = | | = 51 510 (5) ≡ mec meωc − at optical frequenciesω 1 eV/�. As the technical challenge grows with ∼ 2 the intensity, proportional toa 0, laser technology faces still some effort to 2 5 bridge the remaining few orders of magnitude toa 0 mc /�ω 5 10 � → × corresponding to the critical fieldE c. Along side increasing average intensity, fine-tuning the control over the shape of laser pulses proceeds apace. Pulses consisting of only 10 waves are possible today, which translates into a duration hc 10λ = 10 = 12.5µm = 40c fs. (6) 1 eV Contrast at the front of the laser pulse controls much of the pulse-particle interaction since for these extraordinarily high intensities the prepulse can already alter in a dramatic way the state with which the main laser pulse interacts. Developments such as relativistic plasma mirrors [15] have produced dramatic improvements in contrast, approaching the physical limit on the rise of the potential, λ/4, imposed by the wavelength of light used. Since force is proportional to the gradient of the potential, this limit implies that for an electron to achieve unit acceleration in a laser field, the wavelength — as observed by electron — must be decreased. 87 Strong Field Physics: Probing Critical Acceleration and Inertia . . . 2767

2.2. Laser pulse-electron collider To confront a particle with unit force, we therefore propose to take advantage of the Lorentz transformation of the electromagnetic ﬁeld, in a manner analogous to the particle colliders such as LHC and RHIC. An electron with momentum �p= γm�v incident on a laser pulse with the momentum (wave-vector) �k will see the ﬁeld Doppler-shifted in its rest frame

ω ω � =γ ω+ �v �k . (7) → · � � The force and acceleration achieved in the collision is thus multiplied by 2γ for nearly opposite momenta. The diﬃculty in arranging the collision ∼ also increases withγ, because the pulse appears contracted

1 1 l� =l (γ cosθ) − , l� =l (γ sinθ) − (8) � � ⊥ ⊥ in the longitudinal and transverse directions, respectively. Some of the technical challenges have been discussed in the literature and a laser pulse-electron storage ring, analogous to those in hadron–hadron or electron–electron machines, has been proposed [16]. The recent advances in laser technology have reinvigorated investigation of how to observe the QED nonlinearities arising asE E [17]. Yet even before laser intensities → c climbed rapidly in the past decade, the capability to perform experiments of this sort was demonstrated at SLAC [18, 19], and thus we know that it is possible to arrange a head-on collision of a relativistic 47 GeV electron beam with a high intensity focused light pulse. The peak normalized laser intensity attained 15 years ago was

a0 = 0.4 (9)

and with 46.6 GeV incident electrons consequently saw an acceleration

˙u 0.073 [m ]. (10) ≤ e While this magnitude falls short of critical˙u 1, the experiment suc- → cessfully observed multi-photon phenomena predicted by strong field QED, including evidence for nonlinear Compton scattering and electron–positron pairs produced via a Breit–Wheeler process. Repeating the SLAC experiment with 15 years’ improved off-the-shelf laser technology would be a relatively easy step and would lead to study of physics arising at critical acceleration. The center-of-mass energy available in a laser pulse–electron collision surpasses that commonly available in other major probes of strong field physics, including hadron–hadron collisions studied at RHIC (see Section 4). 88 2768 L. Labun, J. Rafelski

A laser pulse consisting of nω 10 22 1 eV photons = 1.6 kJ (11) ≥ × is a coherent object of mass-energy 1010 times larger than the TeV pro- tons available at the LHC. Today it is possible to imagine a coherently- synchronized 1000-beam system (NIF has 192) with individual beam intensities 10 times higher. Such a futuristic hyper-laser would still comprise only a percent-fraction of the Planck mass 28 6 MPl = 1.22 10 eV� 10 (1.6 kJ laser pulse). (12) × × Although achieving Planck energy remains far away, coherent light pulses appear much closer to this goal than any other known physical system and certainly oﬀer much better hope than hadron–hadron colliders [20].

3. Acceleration and radiation reaction 3.1. Planck scale acceleration We have set the stage to argue that experiments can be performed today at unit acceleration, which is a natural scale to expect new physics. Recall that “unit value” quantities defined by settingG=�=c=1 are Planck’s units that arise from the closed, natural dimensioned system created by the introduction of� (see the last page in his work [21]). Unit acceleration in natural units˙u 1[m], thus presents the Planck scale of the “theory of → acceleration”. In order to avoid fruitless discussions of the matter, we do not call critical acceleration “Planck acceleration” since this clearly was not on the list he presented. Acceleration, viewed as a change in 4-velocity in a given field, is different for different mass particles. The condition Eq. (3) is made universal normalizing by the mass, introducing the critical specific acceleration ℵ 3 ˙u c 23 m = 1.37 10 2 . (13) ℵ ≡ m → � × s kg

Newtonian gravity near a body of Planck massM=M Pl �c/G exhibits ≡ unit acceleration at Planck distanceL Pl =�/(M Plc), � ˙v G c3 G = 2 atr=L Pl . (14) ℵ ≡ MPl r → � Creating elementary, or coherent, Planck mass (energy) scale objects and probing the gravity-related Planck scale directly will remain a challenge for some time to come. However, the Equivalence Principle assures that we access the same physics when realizing critical acceleration Eq. (13) in the context of other (electromagnetic and/or strong) interactions. 89 Strong Field Physics: Probing Critical Acceleration and Inertia . . . 2769

3.2. Is electromagnetism a consistent theory? Consider a localized electron incident on a largen laser ﬁeld. The classical character of the initial state leads one to believe that classical dynamics as encoded by the Maxwell equations and the Lorentz force Eq. (2) should apply. However, it has long been recognized that the system of classical equations of motion is incomplete [9]. Speciﬁcally, the inhomogeneous Maxwell equation βα α ∂βF =j (15) leads to radiation by an accelerated charge at the rate

dE 2 duα 2 = = e2 (16) R dt − 3 dτ lab � � (see for example [22]). The inertial reaction of the particle to this energy- momentum loss is not included in the Lorentz force Eq. (2). This observation leads to the iterative procedure: 1. Solve Lorentz force for space-time path of particle in prescribed external field. 2. Compute emanating radiation fields according to Maxwell equations. 3. Correct field configuration and return to step 1. Efforts to close the iteration loop into a single dynamical radiation- reaction incorporating equation have occupied authors from Lorentz through the present. The earliest effort, the Lorentz–Abraham–Dirac equation [23]

m˙uα = eF αβu + mτ üα u β üuα , (17) − β 0 − β � � was shown by Dirac to follow from covariant energy-momentum conservation in the action Eq. (18) [9]. However, Eq. (17) fails to be a predictive equation of motion, because it contains third derivatives of positionü, requiring a third boundary condition to specify a solution. While the extra condition is able to eliminate unstable “run-away” solutions of Eq. (17), its utilization is generally regarded as an unphysical violation of causality. It is of importance to realize that in principle the consistency of classical electromagnetism is not assured, as there are several independent dynamical components; the total action is the sum of electromagnetic, matter, and interaction components 1 mc = d4x F 2 + dτ u2 1 +q dxα(τ)A . (18) I − 4 2 − α � � � path � � path 90 2770 L. Labun, J. Rafelski

The latter two integrations follow the space-time path of the particle with chargeq(= e for the electron). The equations of motion for the to- − tal system are usually obtained separately for the field and the particle: Maxwell’s Eq. (15) by variation of the 1st and 3rd terms, and the Lorentz force Eq. (2) by variation of the 2nd and 3rd terms. Combining the two into one dynamical equation Eq. (17) exhibits their inconsistency, since the derivation of Eq. (17) appears to be physically sound, yet does not produce a physical theory, except in the limit when the acceleration is small˙u m. In � this case, the radiation rate is also small. In particular, since (using natural units) mE 2 ˙u mE/E and e 2( ˙u2) =e 2 , (19) ∼ c R∼ E � c � the effect of radiation reaction can be treated perturbatively as long as E/Ec <1. A rigorous procedure for carrying this expansion to arbitrary order has been recently derived [24]. However, a deeper understanding of the physics is necessary for efficient predictive control over dynamics in the unit-acceleration regime.

3.3. Eﬀorts to improve the classical theory

Many eﬀorts in the intervening years have attempted other solutions (see [5] for a list), most recently even including some quantum eﬀects [25]. Of these, the equation set forth in [26] has received the most attention, because it implements the iteration procedure described above into a perturbative expansion by replacing

d e üα F αβu . (20) → dτ −m β � e � The resulting nonlinear equation, confirmed by the expansion in [24] e m˙uα = eF αβu eτ F αβu uγ F αβF uγ F βγF uδu uα , − β − 0 ,γ β − m βγ − γδ β 2 � � � � 2 e /4πε0 24 τ = = 6.26 10 − s (21) 0 3 mc3 × can be solved analytically in several simple but useful prescribed fields, including notably a transverse “laser” electromagnetic wave [27,28]. This solution of Eq. (21) exhibits several features important for potential experiment. It displays damping of the electron motion, as would be expected from the dissipative nature of radiation, and consequently predicts less total integrated radiation emission than the uncorrected Lorentz force. 91 Strong Field Physics: Probing Critical Acceleration and Inertia . . . 2771

This prediction, among others, demonstrate that the effects of radiation reaction will be easily discernible in the radiation emission and trajectories of the electrons. In the case of a head-on collision between a highly relativistic electron and a high intensity laser pulse, an electron Lorentz factorγ=10 3 and normalized laser intensitya 0 = 100 more than suffice to make radiation reaction effects visible [28]. An important point to recognize is that the trajectories predicted by the Eqs. (17) and (21) are different even when both are considered valid. This distinction arises from the perturbative expansion producing the Landau– Lifshitz expression Eq. (21), which converges more weakly as the radiation correction terms become comparable to the Lorentz force itself. Hence the predictive power of Eq. (21) breaks down approximately as

ωa2γτ 1, (22) 0 0 ∼ whereγ is the Lorentz factor of the incident electron [26, 28, 29]. This condition diﬀers from the expected onset of the relevance of quantum eﬀects

ωa γ m (23) 0 ∼ e corresponding to critical acceleration and the critical ﬁeld strengthE c, Eq. (1), observed in the electron’s rest frame.

3.4. Accessibility of radiation-reaction eﬀects in classical domain

The accessibility of radiation reaction eﬀects within the domain of classical dynamics can be seen directly by considering the invariant spacelike acceleration a2 = ˙uα ˙u (24) − α for an electron colliding with a laser plane wave according to the Lorentz force Eq. (2). We consider an electron of rapidityy, 4-velocity

uα = (coshy,0,0, sinhy) − incident on an oppositely traveling wave

0E 1 0 0 α E1 0 0B 2 u Fαβ = (coshy,0,0, sinhy)  −  − 0 0 0 0  0 B 2 0 0   −   α T  = (0,E1 coshy+B 2 sinhy,0, 0) = (F βαu ) 92 2772 L. Labun, J. Rafelski

and thus

e2 e2 a2 = uαF F βγu = (E coshy+B sinhy) 2 L − m2 αβ γ m2 1 2 2 � 2 � 2 2 2 e E1 +B2 2y E1 B2 2y E1 B2 2 2y = e + − e− + − (a ω) e . m2 2 2 2 → 0 �� (25)

In the last, we use again the dimensionless laser amplitudea 0 Eq. (5). With foreseeable technology settingω 10 5 m anda 103, an incident ∼ − 0 ∼ electron must have rapidityy=4.6 for its acceleration according to the Lorentz forcea L to approach unity. The contours in figure 2 display the levels at which radiation reaction effects are important according to Eq. (22) and the onset of quantum effects at unit acceleration, Eq. (23). We allow considerable margin for the quantum effects to become relevant due to the sensitivity of a real experiment to residual matter triggering cascades [4] and frequency effects [30]. The shaded region highlights parameter space where classical dynamics are expected to remain dominant while radiation reaction effects are non-negligible, specifically requiring an improvement over Eq. (21), due to the poor convergence of the Landau–Lifshitz expansion.

. 2 u = 1 0.1 . 2 u = 0.1 2 -2 ω (a ) γ τ = 1 10 0 0 2

) -3

c 10

(E/E -4 10

-5 10

-6 10

0 1 2 3 4 5 6 y

Fig. 2. Domain of interest for electron of rapidityy moving oppositely to a laser 2 ﬁeld of normalized intensity(E/E c) . The Lorentz-force critical (unit) acceleration is the upper solid line. The shaded region delimited by Eq. (23) from below and a <0.1 from above highlights the parameter space where classical dynamics are expected to remain dominant and radiation reaction eﬀects will be non-negligible. 93 Strong Field Physics: Probing Critical Acceleration and Inertia . . . 2773

Radiation reaction must therefore be dealt with also in a regime where classical dynamics are dominant. Investigations such as [28, 29, 31] begin to address the gap in understanding high acceleration dynamics within classical theory. An open question is how much guidance quantum electrodynamics (QED) will offer in this endeavor. There is widespread belief that QED has cured the defects of classical theory, yet there is no firm evidence that this is the case. At this point it has been shown [32, 33] that QED can repro- duce the classical equations of motion to second order in perturbation theory when unusual boundary conditions are imposed. The issues preventing understanding of the relationship of time-reflection invariant QED and a causal classical theory have been recognized often in the past but not resolved, see comments made by Spohn in Section 1 of his recent monograph [34].

4. Elementary and heavy ion high energy collisions 4.1. Critical acceleration and quark-gluon plasma formation An interesting early precursor to the current discussion of high acceleration phenomena is the hypothesis by Barshay and Troost [35] that achieving high acceleration in elementary interactions could explain the thermal characteristics of the multiparticle production phenomena in terms of the Hawking–Unruh effect (see Section 5). This idea has been further elaborated in recent years by Satz and collaborators [36,37]. We do not take the perspective that all multiparticle production is due to strong acceleration effects: there are excellent reasons within the realm of strongly interacting matter to expect the emergence of a thermalized state in heavy ion collisions. However, in high energy particle collisions the strong acceleration phenomena could contribute to establishing thermal equilibrium of particle yields. To illustrate this line of thought, we first show that some matter partic- ipating in heavy ion reactions does achieve critical acceleration due to the strong stopping of quarks that has been reported. While the larger fraction of each nucleus passes through without large loss of rapidity (energy), a significant fraction of valence quarks, at the level of 5% of both projectile � and target are stopped in the center of momentum frame [38]. This means that the scaling domain, expected at ultra-high energy [39], has not been reached. For initial momenta of the componentsM i pµ = (M coshy ,0,0, M sinhy ), (26) i p ± i p wherey p the rapidity of the incident beam, the accelerationa, Eq. (24), required to stop a parton within a proper timeΔτ is

yp a=˙y . (27) � Δτ 94 2774 L. Labun, J. Rafelski yp = 5.4 at RHIC andy p = 2.9 at CERN-SPS, so for a constituent quark of massM M /3 310 MeV to undergo critical acceleration, it must be i � N � stopped withinΔτ <3.4 fm/c orΔτ <1.8 fm/c, respectively at RHIC and SPS. In fact, for the SPS experiments with a 30 GeV beam incident on a ﬁxed target,Δτ <1.3 fm/c. This time scale is comparable to the “natural” quark-gluon plasma formation timeτ 0 = 0.5–1 fm/c [39].

4.2. Anomalous soft photon production In addition to approaching critical acceleration with respect to quantum chromodynamic forces, high energy particle collisions reveal the possibility of incomplete understanding of radiation reaction in the high energy regime. Many of the colliding particles and in particular the strongly-interacting quarks carry electromagnetic charge. Their high accelerations during the collision generate gluons and photons. The eﬀect of radiation reaction due to this radiation accentuates the stopping power and augments the gluon and photon radiation yield. Such considerations could explain the puzzling excess of soft photon production above perturbative QED expectation in many experiments. Wong oﬀers a comprehensive summary of the phenomena in need of an explanation [40]. Anomalous soft photon production in elementary high energy interactions is observed universally in conjunction with the production of hadrons, mostly mesons: inK +p reactions [41,42], inπ +p reactions [42], in + π−p [43–45], inpp collisions [46], in high-energye –e− annihilations through Z0 hadronic decay [47–50]. The main features of the anomalous soft photon production are summarized as follows:

1. Anomalous soft photons are produced in association with hadron production at high energies. They are absent when there is no hadron production [48].

2. The anomalous soft photon yield is proportional to the hadron yield.

3. The anomalous soft photons carry signiﬁcant transverse momenta, in the range of many tens of MeV/c.

4. The yield of anomalous soft photons increases approximately linearly with the number of neutral or charged produced particles.

Especially the ﬁrst and last feature suggest that photons and gluons are produced together in strong stopping of quarks, with gluons turning into neutral hadrons at hadronization. This corroborates the possibility that the quark stopping is driven by eﬀective radiation reaction forces, akin to 95 Strong Field Physics: Probing Critical Acceleration and Inertia . . . 2775

those we discussed for the case of the classical electromagnetism. To prove this conjecture will require a more complete study of the radiation reaction phenomena at the quark level.

5. Quantum vacuum and acceleration 5.1. Vacuum instability As noted by Sauter [10], Euler and Heisenberg [51], and Schwinger [11], strong electric fields are susceptible to conversion into electron–positron pairs. The field strengthE c, Eq. (1) is the non-perturbative scale of the barrier to vacuum decay. The materialization is global and very rapid when the field achieves critical strength — i.e. when electrons and positrons ex- perience above-critical acceleration. A practical description of the decay is as a semi-classical tunneling process, which becomes non-negligible as the potential becomes strong enough to accelerate an electron across the gap in the Fermi spectrum

eE Slope of potential 2m2c3 = Gap width = 2mc2 orE . m/�c Scale of wavefunction ∼ ∼ e�

Up to a factor 2, this is just the condition for critical ﬁeld strength Eq. (1) and unit acceleration Eq. (3). Two typical timescales are associated with the lifespan of the quantum vacuum state with applied ﬁeld.

1. The total probability of decay via pair-creation of the zero-particle state in the presence of a given ﬁeld strengthE

2 (eE) ∞ 1 πEc Γ= exp n (28) 4π3 n2 − E n=1 � � � is the “width” of the with-field state and hence the imaginary part of an effective potential. This rate does not illuminate what happens to the state at strong field but simply conveys the message how quickly the instability takes hold. This is the dashed (red) line in figure 3.

2. A physical definition of the persistence of the field is obtained by studying the conversion of field energy into particle pairs [52]. As the critical field is approached, the vacuum materializes at a rate

1 1 d um τ − = � � , (29) uf dt 96 2776 L. Labun, J. Rafelski

12 10 9 10 −1 ω − electron Compton time 6 e 10 40 fs 3 3 −1 10 (L Γ/2 ) − inverse vacuum width τ − 1 materialization time -3 10 -6 10 t [ s ] -9 10 -12 10 -15 10 -18 10 -21 10 0.1 1 10 E / Ec Fig. 3. The lifespan (inverse decay half-width) of the with-field quantum vacuum state (dashed line) and the energy content of the field (solid line) as function of the externally applied field strengthE/E c. The time length of presently available laser pulses (40 fs, short-dashed horizontal line) and the inverse Compton frequency of the electron (long-dashed horizontal line) provide reference for the time scale.

whereu f is the energy density available in the electromagnetic ﬁeld and ∞ d um 2seE 2 πM 2 /E � � = 2 dM 2M e− ⊥ (30) dt 4π ⊥ ⊥ m�e

is the expected rate at which energy is converted into electron–positron pairs. The result is shown as the solid line in ﬁgure 3.

For comparison the inverse electron Compton frequency (long-dashed horizontal line) and the typical laser pulse time, 40 fs (short-dashed horizontal line) are also shown in figure 3. Interestingly, we note that the laser pulse lifespan is of the same magnitude as the pulse length already at E=0.3E c. On the other hand, the materialization of the field energy is not as fast as the Compton frequency. This result is due to a factor α/π 1 difference betweenω e− and the dimensionful coefficient inτ and can be interpreted to mean that the weakness of the QED coupling implies that it is not necessary to implement back-reaction of produced pairs on the applied field, though such an approach is certainly required for a fully consistent description [53]. 97 Strong Field Physics: Probing Critical Acceleration and Inertia . . . 2777

5.2. Subcritical pair production The analytic completion of the imaginary part Eq. (28) is the eﬀective potential derived by Euler, Heisenberg and Schwinger

∞ 1 2s dt eEt m2t V =i Tr ln G− [F ] = 1 e− . (31) eﬀ ext − 16π2 t3+� tan eEt − � � � � � 0

The Green’s function in the external ﬁeldG[F ext] appearing in Eq. (31) can be decomposed as a sum over eigenstates in the external ﬁeld

G(x, x�) = cλ Tˆ ψλ(x)ψ¯λ(x�), (32) λ= pµ,σ �{ }

whereψ λ is an eigenfunction of the with-ﬁeld Dirac equation, Tˆ the time- ordering operator, andc λ a normalization constant. This propagator has been explicitly computed [54,55] and was used for the baseline predictions for the electron-laser pulse collisions at SLAC [18,19]. As noted, the experiment remained below the level at which quantum eﬀects are important and critical acceleration is attained, and relatively good agreement seen between the semi-classical evaluation ofG(x, x �) utilized and the predominantly classical conditions in the setup.

5.3. Accelerated vacuum and Hawking–Unruh radiation

The decomposition of the Green’s function Eq. (32) shows howV eff integrates the effect of the external field on the single particle states and thus represents an “accelerated” vacuum state. The natural next question is whether this description of the vacuum accelerated by the external field is consistent with the description of the quantum vacuum according an accelerated observer. Unruh showed [56] that a detector (a scalar particle confined to a box) undergoing constant accelerationa is excited in a Planckian spectrum with Bose statistics and temperature a T = . (33) HU 2π

Subsequent work has re-derived this result from many diﬀerent approaches, and in every case, the temperature agrees with the Hawking–Unruh temperature and the statistics match those of the un-accelerated quantum ﬁeld [57]. 98 2778 L. Labun, J. Rafelski

In contrast, the eﬀective action Eq. (31) has a Planckian representation

4 ∞ m TEH 2 2 ω/T V = 2 ln ω /m 1+ i� ln 1 e − EH d(ω/m), eﬀ (4π)2 m s − − � 0 � � � � eE a T = = , (34) EH πm π

in which the temperature is twice the Hawking–Unruh temperature,T EH = 2THU, and the statistics of the distribution are inverted, displaying a Bose- like ( ) [58]. Computing the effective potentialV for a spin-less “electron” − eff results in the same temperatureT EH and a “wrong” Fermi-like sign in the distribution [59]. This situation is summarized in Table I and reveals puzzles that remain in the non-perturbative predictions of the quantum theory at the critical scaleE E . → c TABLE I The thermal characteristics of acceleration radiation contrasted with those found for a constant electric field.

Accelerated observer External field–accelerated vacuum detector accelerated against electron Fermi sea states flat-space vacuum at constant accelerationa= eE/m detector response function sum negative energy states thermal excitation spectrum effective potential → → THU = a/2π TEH = a/π statistics match statistics inversion (boson boson) (boson fermion) (fermion �→ fermion) (fermion �→ boson) �→ �→

5.4. The quantum vacuum frame and the æther One only appreciates the challenge of bringing together the discussion of acceleration and the quantum vacuum recollecting the conflicting descriptions of accelerations provided by general relativity and quantum theories. First, one should bear in mind that in the absence of quantum theory, extended matter objects are hard to imagine. Without a finite size, point-like particles fall freely in gravitational fields, and there is no acceleration to be discussed. Resistance to gravitational free fall is in essence only possible since quantum atoms have finite size and many atoms come together to form macroscopic objects that resist the pull of moderate gravitational forces. In turn, our ability to assemble devices producing critical acceleration thus depends on the quantum nature of matter and radiation. We may 99 Strong Field Physics: Probing Critical Acceleration and Inertia . . . 2779 therefore expect that the study of physical phenomena at unit acceleration is likely to advance our understanding of the difficulties in uniting gravitational and quantum theories, since their inconsistency is accentuated and the equivalence principle challenged. Further, quantum theory contains an universal inertial state, which is the global, lowest-energy (i.e. quantum vacuum) state. Textbook treatments of the classical limit of quantum theory do not involve this reference frame. However, in presence of strong acceleration a more refined classical limit could be required which generates a modified Lorentz equation, wherein acceleration refers to the vacuum state as the universal inertial frame. A promising method to derive a classical limit that includes vacuum dynamics as well as back-reaction has been developed within the relativistic Wigner function formulation [60]. Within this framework of phase space functions there seems to be a natural opportunity to derive particle dynamics with the vacuum state present as a reference frame identifying classical acceleration. The classical dynamical equations in phase space arise naturally [61]; the classical limit requires coarse-graining and, so far, back-reaction effects have not been considered. The effective forces obtained were at this level of discussion identical to the known classical Lorentz and Bargmann–Michel–Telegdi equations. Noteworthy in the above discussion is that the quantum vacuum as- sumes the role of the relativistically invariant æther, the intangible carrier of physical law that Einstein proposed around 1920 [62], reexamining his earlier criticism of the æther hypothesis. This modern æther (the quantum vacuum) provides a natural preferred inertial frame perhaps capable of resolving the debate inspired by Mach and Einstein over how to define inertia. The challenge of understanding inertia may thus require the inclusion quantum vacuum structure. Remarkably, the current paradigms of quantum field theory invoke quantum vacuum structure as the preeminent source of the definition of the inertial mass of all particles, from electroweak symmetry breaking and minimal Higgs coupling to color confinement. On the other hand, reconnecting the presence of the quantum vacuum to the classical realm remains difficult with the classical limit of the quantum theory eluding full understanding, and in particular not referring acceleration to the presence of the vacuum state.

6. Conclusion Taken in isolation, many of the physics topics discussed here are well- known, if not in every case perfectly understood. Our purpose has been to unite apparently disparate phenomena and in-principle considerations under the common theme of high-acceleration physics and point to a few 100 2780 L. Labun, J. Rafelski new insights, some at present still hypothetical, but all accessible to in depth study via experimental technologies either immediately available or presently in development. Laser technology nears the capability to perform finely-controlled experiments in which charged particles attain and considerably exceed the critical acceleration˙u 1[m]. Since the quantum vacuum structure of electrody- → namics is simpler than that of chromodynamics, we expect that this context will provide cleaner experimental and theoretical access to the physics of radiation reaction and dynamics in the unit-acceleration regime. On the other hand, the lesson of heavy ion collisions and multi particle production may be that the phenomena are fundamentally linked by critical acceleration. To pose, much less answer the question of defining acceleration, the dynamical theory of matter and radiation must at least incorporate the inertial response of the charge to its own radiation, a self-consistency not yet obviously in hand for classical or quantum electrodynamics. Thus, while investigation of particle production and quantum vacuum structure may offer guidance in the quest to understand matter and inertia, the associated challenges in the classical domain, and in particular radiation reaction, must be independently addressed. The ongoing and forthcoming experimental efforts will without doubt lead to a renaissance in strong field QED and QCD physics. The array of topics covered here and the connection through the Equivalence Principle to the gravitational theory and Planck scale highlights both difficulties and opportunities provided by strong fields and critical acceleration to understand the present theoretical framework encompassing quantum and gravitational theories.

This work was supported by the grant from the U.S. Department of Energy, DE-FG02-04ER41318.

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APPENDIX F

Vacuum Structure and Dark Energy

Vacuum Structure and Dark Energy Labun, L. and Rafelski, J. International Journal of Modern Physics D Vol. 19, No. 14 (2010) 2299-2304 doi: 10.1142/S0218271810018384 arXiv:1011.3497 [hep-ph]

Copyright (2010) World Scientiﬁc Publishing Company Reprint permission granted for post-print reproduction under World Scientiﬁc Author’s Rights: http://www.worldscinet.com/authors/authorrights.shtml 105

Vacuum Structure and Dark Energy1

Lance Labun and Johann Rafelski Department of Physics, The University of Arizona, Tucson, AZ 85721, USA

Abstract

We consider that the universe is trapped in an excited vacuum state and the resulting excitation energy provides the observed dark energy. We explore the conditions under which this situation can arise from physics already known. Considering the example of how macroscopic QED ﬁelds alter the vacuum structure, we ﬁnd that the energy scale 1 meV — 1 eV is particularly interesting. We discuss how dark energy of this form is accessible to laboratory experiments.

An excited quantum vacuum state known as the quark-gluon plasma predominated in 12 the early universe when temperatures exceededT h ∼ 160 MeV= 1.8×10 K and the age of the Universe was less than 25µs. In the expansion and cooling the quark-filled Universe transformed belowT h into the current matter state with nearly equal abundances of hadrons and their antiparticles. Drops of excited quark-deconfined state can be trapped inside hadrons as depicted first by the MIT quark-bag model [1]. In the subsequent evolution, matter and antimatter annihilated, neutrinos froze out, primordial nucleosynthesis took place and ultimately CMB radiation decoupled—all processes well-known in principle. We consider here the possibility that the universe relaxed into one of the many nearly-degenerate metastable vacuum states arising within e.g. the framework of strong interactions, QCD. This first exploration is carried out in the more manageable environment of QED in the presence of macroscopic applied fields, and our finding determines the scales and is suggestive of beyond the standard model physics. Since the onset of interest in the deconfined quark-gluon plasma phase, it has been recognized that the early Universe would have as a significant component in the energy balance the cosmological constant-type term usually referred to asB due to thebag model [2]: a homogeneous distribution of vacuum energy. The inward pressure of the fluctuations arises since the excited vacuum is necessarily unstable, ready to collapse—the excited vacuum is ‘pulling’ inward on the surrounding True Vacuum|TV�, with total collapse prevented by conservation of quantum charges such as baryon number and/or electrical charge. At the time quarks roamed free in the Universe, the effective dark energy (=B� (170MeV) 4) was larger by 43 orders of magnitude than it is today. While the QCD vacuum state is unique, it is ‘surrounded’ on the scale of QCD by a practical continuum of degenerate states, related to the question how strong interactions preserve the CP symmetry [3]. This is an unresolved issue, and for our purposes it suffices to observe that the Universe may relax from the highly excited deconfined state to one of the many available metastable QCD vacuum states that are (measured on QCD scale) very near to the true ground state. As the Universe continues to evolve, transitions between these states can occur spontaneously, especially at higher temperatures releasing energy [4], adding heat to the the thermal bath, and thus improving the homogeneity of the CMB

1Recognized for Honorable Mention in the Gravity Research Foundation 2010 Awards for Essays on Gravitation, to appear in Int. J. Mod. Phys. D 106

radiation. In the absence of physics strongly breaking the degeneracy, the Universe at a sufficiently low ambient temperature may freeze into a low-lying excited state which has a lifespan greater than cosmological times. The QCD vacuum structure thus offers a framework near to the known physics that leads naturally to consideration of dark energy and the observed cosmological acceleration [5, 6]. However, we must remember that a full understanding of the QCD vacuum remains elusive. For this reason and in order to gain more quantitative grasp on the hypothesis that the Universe is in an excited vacuum state, the present investigation is carried out in the more manageable environment of QED in the presence of macroscopic applied fields. We explore the scales relevant if dark energy arises as the energy of a vacuum excited by a universe-spanning constant electric and/or magnetic field. In this situation, we find a simulacrum to dark energy [7] and we can study the metastability and energy content of the excited vacuum. The novel element explored here is the magnitude of mass scale required to explain dark energy in this context. The vacuum filled with electromagnetic field is a metastable state because the conversion of field energy into particle-antiparticle pairs is possible. The process is very slow except in very strong fields m2 E=β −1 , β−1 →1 (1) e introducingβ −1 as an inverse Hawking-Unruh temperature parameter. 2m is the energy stability gap, 2/m the width of the tunneling barrier, ande the coupling charge.β −1 �0.1 suffices to induce just one decay event in a macroscopic volume, while full collapse of the vacuum ensues forβ −1 → 1 [8]. As dark energy is a long wavelength phenomenon, we study the long-wavelength limit of QED in prescribed external fields of arbitrary strength with small values of the charge e2/4π2 = α/π. The field is constant and homogeneous, filling our model universe and hence the relevant object to study is the effective Euler-Heisenberg-Schwinger QED action [9]. It can be written in the form

4 S P Veﬀ ≡ −S+m feﬀ , ; (2) �m4 m4 �

1 αβ wheref eﬀ , found in textbooks, is a highly nonlinear function of its argumentsS= 4 F Fαβ 1 ∗ αβ andP= 4 F Fαβ. Well-understood charge renormalization terms also enter Eq. (2) and are omitted here. The total energy-momentum tensor

µν ∂Veff µν µλ ν µν ∂Veff ∂Veff T = − (g S −F F )−g Veff − S − P (3) � ∂S � λ � ∂S ∂P � exhibits the relation of the energy-momentum trace to the scalem

µ ∂Veff ∂Veff dfeff T =−4 Veff − S − P =−m (4) µ � ∂S ∂P � dm

The Euler-Heisenberg-Schwinger calculation exhibits how the electromagnetic ﬁeld is responsible for the ‘false vacuum’ by pushing the electron-positron ﬂuctuations away from 107

|TV�. The augmented energy of these fluctuations implies the noted instability of the with- field vacuum and leads to the dark energy T µ λ→ µ . (5) 4 µ Classical fields generatingT µ and satisfying the long wavelength approximation are created regularly in experiments. The reason they do not usually provide insight on dark energy, vacuum structure and instability is most clear from the timescaleτ corresponding to the lifespan of with-field vacuum state.τ being exponential inβ generally far exceeds the age of the universe: for a (QED) electric field very strong by laboratory standards,∼ 10 10 V/m, the associated timescale is 1054 years. This field is chosen as our example because it generates an energy-momentum trace comparable to the dark energy density

µ −10 3 4 Tµ (E=32.4 GV/m)=6.09×10 J/m = (2.3 meV) (6)

Setting the scalem at the electron mass (0.5 MeV) is the cause in this example of the discordant result that a field gargantuan in laboratory terms is required to induce a small vacuum energy. We can ask what scalem admits a field that has a lifetime exceeding the age of the universe and hence can persist up to the present while generating the observed dark energy. No optimization is attempted here and in particular the coupling is left fixed at the QEDα, but in the figure, we see that the strength of the field must remain below β−1 �.04 for the excited state to live as long as the universe without decaying. Up to factors of order unity,m is found in the range 1 meV – 1 eV which is suggestive that dark energy is a phenomenon buried in depths of and beyond the standard model physics, related to the above-mentioned QCD vacuum degeneracy and/or neutrinos and their masses [10]. The non-perturbative maximum in the energy-momentum trace atβ −1 �5.6 and subsequent change in sign are manifestations of the instability of the electric field, resulting from the rapidly increasing imaginary part [7]. We can hope to manipulate soon this field-induced simulacrum of the dark energy. High intensity pulsed laser experiments will achieve electromagnetic fields closer to the 2 −1 µ critical scaleE→m /e, that isβ → 1, even whenm=m e and hence allow the studyT µ and the vacuum structure. The scale we invoke 0.001

36 10 36 1 meV 30 1 meV 27 1 eV 10 1 eV 1 MeV 24 1 MeV 18 10 λ ] 18 d

9 ] 10 4 univ univ 12 0 10

µ µ 6

-9 [ meV T

log [ τ / t 10

-18 1

-27 -6 10

-36 -12 10 0.01 0.1 1 10 100 10-8 10-6 10-4 0.01 1 100 −1 −1 β β

Figure 1: At left, the lifetimes of electric fieldsE=β −1m2/e in terms of the age of the universet univ for various values of the mass scale. Note that form = 1 meV, a field of .25 V/m (β=0.05) has a lifetime equal to the age of universe, and any smaller field will survive into the present era without decaying. At right, the field induced energy-momentum trace with the dark energy density at the horizontal line for comparison. radiation, which is characterized by the Hawking-Unruh temperature

a eE −1 THU = , wherea= = mβ (8) 2π m is the acceleration of an electron in an electric fieldE. One recognizes that fields causing acceleration at the scale of unity connect to an unstable effective action describable in the framework of the Hawking-Unruh temperature, though some technical nuances still need to be resolved [12]. High intensity pulsed laser-electron collisions in which a relativistic charged particle is stopped within one wavelength generate critical acceleration and thus can test the limit of the current concept of interactions. Other experimental approaches include LHC heavy ion reactions and in the foreseeable future laser pulse-on-laser pulse collisions. By probing consequences of the force scalem 2/e, these experiments also access the nature of the excited vacuum. We conclude that resolving vacuum structure in strong fields is a natural and well- defined experimental and theoretical context with results that may bear on the mystery of dark energy. Its consequences are available to laboratory study by exploring strong fields µ and the effects of the induced energy-momentum traceT µ and critical acceleration. The ultimate test for an excited vacuum state generating the dark energy is the observation of a release of the excitation energy associated with the vacuum transition. This process has recently been investigated for the de Sitter geometry, which is the gravitational analog of the pair-creating electric field [13]. In view of their ability to affect vacuum structure, it can be hoped that strong applied QED fields could catalyze decay of the ‘false’ vacuum. 109

Acknowledgments This work was supported by a grant from the U.S. Department of Energy DE-FG02- 04ER41318. References

[1] J. F. Donoghue, E. Golowich and B. R. Holstein, Dynamics of the Standard Model, (University Press, Cambridge, 1992).

[2] J. Kapusta, B. M¨uller and J. Rafelski, Quark-Gluon Plasma: Theoretical Foundations (Elsevier, Amsterdam, 2003).

[3] P. Ramond, Journeys Beyond The Standard Model (Perseus Books, Reading, 1999).

[4] J. Rafelski, L. Labun, Y. Hadad and P. Chen, in Tenth Workshop on Non- Perturbative Quantum Chromodynamics (SLAC eConf Proceedings, 2009) C0906083: http://www.slac.stanford.edu/econf/C0906083/.

[5] J. Frieman, M. Turner and D. Huterer, Ann. Rev. Astron. Astrophys. 46 (2008) 385.

[6] A. Silvestri and M. Trodden, Rept. Prog. Phys. 72 (2009) 096901.

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[9] H. Euler and B. Kockel, Naturwiss. 23 (1935) 246; H. Euler, Ann. Physik. (Berlin) 26 (1936) 398; W. Heisenberg and H. Euler, Z. Phys. 98 (1936) 714, for translation see [arXiv:physics/0605038]; J. S. Schwinger, Phys. Rev. 82 (1951) 664.

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[12] W. Y. Pauchy Hwang and S. P. Kim, Phys. Rev. D 80 (2009) 065004.

[13] A. M. Polyakov, Nucl. Phys. B 834 (2010) 316. 110

APPENDIX G

Spectra of Particles from Laser-Induced Vacuum Decay

Spectra of Particles from Laser-Induced Vacuum Decay Labun, L. and Rafelski, J. Physical Review D84 (2011) 033003 doi: 10.1103/PhysRevD.84.033003 arXiv:1102.5773 [hep-ph]

Copyright (2011) by the American Physical Society Reprint permission granted under APS Copyright Policies: http://publish.aps.org/copyrightFAQ.html#thesis 111 PHYSICAL REVIEW D 84, 033003 (2011) Spectra of particles from laser-induced vacuum decay

Lance Labun and Johann Rafelski Department of Physics, University of Arizona, Tucson, Arizona 85721, USA (Received 28 February 2011; published 5 August 2011) The spectrum of electrons and positrons originating from vacuum decay occurring in the collision of two noncollinear laser pulses is obtained. It displays high energy, highly collimated particle bunches traveling in a direction separate from the laser beams. This result provides an unmistakable signature of the vacuum-decay phenomenon and could suggest a new avenue for development of high energy electron and/or positron beams.

DOI: 10.1103/PhysRevD.84.033003 PACS numbers: 12.20. m, 03.50.De, 11.15.Tk, 42.55. f � �

I. INTRODUCTION the frame of reference in which production occurs. We evaluate for the case of noncollinear colliding lasers, a Quantum electrodynamics implies that, when the poten- field configuration whose local rest frame moves at high tial difference in an electromagnetic field exceeds2mc 2 in rapidity with respect to the laboratory. The advantages of constant [1,2] and nonconstant fields [3], the field-filled this geometry are that (1) nearly monochromatic bunches domain decays by particle emission. Many other effects of high energy particles are produced, and (2) the bunches of the quantum-induced nonlinearity have been studied are boosted in the direction of the Poynting vector over the years, including photon splitting [4,5], the of the combined laser pulses and thus away from each energy-momentum trace [6–8], and vacuum birefringence individual laser pulse. These characteristics constitute a [9,10]. None of these effects has yet been experimentally signal that the pairs originate in vacuum decay, which from observed. the standpoint of discovery potential outweighs the reduc- A plane electromagnetic wave of arbitrary intensity and tion in particle yield as compared to head-on collisions. wavelength composition is absolutely stable against pair Mirroring the practice in elementary particle physics, materialization decay [2], since there is no reference frame the decay rates and products are studied in this work in which massive decay products can appear. Colliding in terms of Lorentz invariants, keeping separate the laser pulses can form a field configuration that does have particle production process from the reference frame a preferred reference frame: the frame in which the local determination. field momentum density (the Poynting vector) vanishes defines the local rest frame in which decay products can materialize. The field energy present in this frame is the II. RAPIDITY OF MOVING REST FRAME amount available for materialization [11]. In the laboratory frame characterized by a 4-velocity In fact, any electromagnetic field configuration having K ~ K KJ u 1; 0 , the 4-momentum density isp T uJ us- an identifiable reference frame can decay, and will decay ¼ ð Þ KJ ¼ for any positive value of the invariant ing the energy-momentum tensorT . The energy density p0 and the Poynting momentum densityp i of an electro- d2 S2 P 2 S>0; (1) magnetic field configuration are ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ � 2 2 K 2 2 E~ B~ T S 1 B~ E~ ;P E~ B:~ (2) p0 " þ K ; pi " E~ B~ i : S:~ (3) ¼ 2ð � Þ ¼ � ¼ 2 þ 4 ¼ ð � Þ ¼ A previously in-depth studied example is pair production 173 The dielectric shift" 1 and the energy-momentum trace by a supercritical Coulomb potential near aZ> nu- K � cleus [12]. There, the bare charge of the at-rest nucleus is TK vanish in classical electromagnetism and acquire non- screened by the charged vacuum state [13] localized zero field dependent values in QED due to fluctuations in around the nucleus, and a positron escapes. If and when the vacuum [8]. The (square) invariant mass density� 2 u TJ TKJ0 u the supercritical potential dissociates, the localized elec- ¼ J K J0 tron charge from the charged vacuum state could also be can be studied in any frame of reference, in particular in the freed, completing the pair formation process. lab frame or in the frame having S~ 0 where energy ¼ Many studies have addressed pair creation in intense density is rest mass density. We proceed to evaluate mass laser pulses colliding head-on [14–27] (since this offers the density in the lab frame. Using the explicit expression for highest particle yield) or traveling parallel [28–32] (since the electromagnetic energy-momentum tensor, we obtain chirped laser pulses consist of many focused frequency [8,11] components). In this work we show how one obtains spe- �2 : p pK " 2 S2 P 2 TK=4 2: (4) ctra of produced particles in the laboratory by identifying ¼ K ¼ ð þ Þ þ ð K Þ

1550-7998=2011=84(3)=033003(5) 033003-1 � 2011 American Physical Society 112 LANCE LABUN AND JOHANN RAFELSKI PHYSICAL REVIEW D 84, 033003 (2011) For general field configurations,� is a function of space shows the momentum distribution to be dominantly and time and determines the instantaneous, local rapidity Gaussian. Because of the adiabatic! m turn-on of � e yS of the moving rest frame, the field, particles are created at threshold with zero mo- 2 2 ~ ~ 2 ~2 0 2 mentump 0 in the direction of the electric field in the 2 � " E B S p k ¼ cosh yS þ ð2 � Þ 1 2 ð 2Þ ; (5) field’s moving rest frame, and the energyE p of the out- ¼ � ¼ þ � ¼ � going particle depends only on the momentum p~ 2 perpen- which exhibits the dependence on the geometric relation dicular to electric field. The tunneling probability? between the electric and magnetic field vectors. As 2 �Ep=ed 2 2 2 for particles, large electromagnetic momentum density � e � ;Ep p~ m (8) / ¼ ? þ S~ >� in a given frame corresponds to high rapidity. and other pertinent quantities depend on the field invari- j j K cosh 0 0 sinh Note that the choiceu yS; ; ; y S produces antsd 2 pS2 P 2 S E~ 2 andb 2 pS2 P 2 the corresponding result¼ for ð mass density workingÞ in the ¼ þ � !j j ¼ þ þ S B~ 2 whichffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi reduce to the corresponding electricffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and local center of mass frame. magnetic!j j magnitudes in the moving rest frame of the field. An isolated electric field is at rest in the laboratory. In Deriving the spectrum requires knowing the orientation of the presence of any magnetic field along with the electric the field: p~ above is the electron (positron) momentum field, the total field configuration acquires momentum transverse to? the direction of the electric field vector dependent on the angle between the electric and magnetic present in the moving rest frame of the decaying field. field vectors, seen in the nonzero Poynting vector Eq. (3). Note that transforming to the laboratory frame preserves If the magnetic field is orthogonal to the electric field the Gaussian shape. P 0 ~ ~ ( ), a frame exists where one of E; B vanishes and The total number of expected pairs N is obtained by ¼ P�0 is the rest frame. Otherwise, when the frame where integrating Eq. (8) over transverse momentah i and multi- ~ ~ E; B are parallel is the rest frame. plying by total volume and time over which the unstable Consider the example of a linearly polarized plane wave field configuration (E2 B 2 >0) exists [ 34,35]: entering a domain of quasiconstant magnetic field. The � 2 2 largest quasistatic magnetic fields in laboratory are small 1 e bd �b �m2=ed ed �m2=ed N coth e�ð Þ ð Þ e�ð Þ: in comparison to the fields available in pulsed laser sys- h i¼ 8�2 � d � b!0 8�3 VT ! tems, and the ratio of the external field to the laser field (9) strength The rate at which field energy is converted into material K ~ eBext 4 e A e Elaser particles is computed by weighting the momentum integral < 10� ; a0 : j j j j ; (6) a0!m ¼ m ¼ !m with the energy of the particle, giving [11] wherea 0 is the dimensionless laser amplitude and! is d um �m 2 �m2=ed h i d e� : (10) the laser frequency. The resulting superposition of fields dt ’ �2 has a small mass density and a large rapidity. The same The characteristic time for conversion of field energy into choices for field geometry that provide a largey S reduce the magnitude of� due to the finite total energy density particles usually far exceeds the laboratory duration of the * ~ being apportioned between rest mass density and momen- field. Ford 0:1 Ec however, pair creation becomes copious and thej lifetimej approaches the time scale of the tum density by coshyS. For a present day high-intensity laser with! 1 eV and laser pulses composing the field. Consequently, the ’ fraction of field energy converted into particles becomes a0 100 and an external magnetic field of 1 T aligned with¼ the laser polarization, the resultant mass density large and detailed treatment of backreaction is necessary � x; t and rapidityy x; t at each position and time are [36,37]. ð Þ ð Þ MeV & 8 * � x; t 5 10 3 ; y x; t 14: (7) IV. COLLIDING LASER PULSES ð Þ � nm ð Þ AND PARTICLE SPECTRA This� is 14 orders of magnitude smaller than the scale set - 3 ~ 2 ~ As the strongest available electromagnetic fields are by the electron0:511 MeV=� e eEc , where Ec m2c3=eℏ 1:3 10 18 V=m 4:4¼ð 10 Þ9 cT, andj j too ¼ today created with pulsed lasers, colliding pulses offer small to generate¼ � a significant¼ number� of electron-positron the best means to maximized and create unstable field pairs. configurations. Judicious choice of collision geometry cre- ates a region where both the rapidityy and the electric invariantd are large. Figure 1 illustrates two linearly III. MATERIALIZATION OF polarized laser waves k~w; k~s (subscripts w for ‘‘weak’’ ELECTRON-POSITRON PAIRS and s for ‘‘strong’’), converging at an angle� with electric Decay of the unstable field into an electron-positron pair field vectors E~s; E~w aligned in thex direction. The wave k is treated as a semiclassical tunneling process [33], which vectorsk w; ks define thex ;1-x ;2 collision plane, in which ? ?

033003-2 113 SPECTRA OF PARTICLES FROM LASER-INDUCED... PHYSICAL REVIEW D 84, 033003 (2011) x 1025 y 12 S

E +E 20 ws 10 10

ks 8 1015

θ x ,2 6 r =0.1 10 r =1 r =0.01 10 δ 4 Stot 105 kw 2 x ,1

0 FIG. 1 (color online). Schematic of two converging linearly 0 10 π/8π/4π/8 π/23 polarized laser waves, k~s; k~w, with E~s; E~w the corresponding electric field vectors. S~tot is the Poynting vector of the resultant θ field, deflected an angle� by the second laser pulse. FIG. 2 (color online). The rapidityy S of the moving rest frame the collision angle� varies between exactly collinear laser (left scale) and expected number of pairs (right scale, lines with arrows) forr 1;0:1;0:01 at fixeda s !s=m 1. pulses� 0 and a head-on collision� �. ¼ 0 ¼ ¼ ¼ The Poynting momentum of the total field S~tot is determined by the net magnetic field: the vector sum of the 2 2 s s 2 b 0; ed 2r 1 cos� a 0! m : (12) lasers’ magnetic fields deflects S~tot away from the beam ¼ ð Þ ¼ ð � Þð Þ line of the strong laser by an angle� given by We consider three cases: equal intensityr 1, and ‘‘weak ¼ w w deflections’’r 0:1, 0:01. The rapidityy S is shown on r sin� a0 ! tan� ;r 1: (11) the left axis in¼ Fig.¼2 with corresponding three lines merg- ¼ 1 r cos� ¼ as !s � þ 0 ing at small� at largey (from top to bottom: smallr to r 1). For the collision of equal intensity pulsesr 1, ~ Stot is in the plane normal to the electric field (the collision the¼ rapidity goes to zero approximately linearly as�¼ �, plane) also when considered in the moving rest frame of reflecting that standing waves created by counterpropagat-! the field. p~ of produced particles is in this collision plane. ? ing square laser pulses are at rest in the laboratory. In the Upon boosting by rapidityy S to the lab frame, the Gaussian asymmetricr<1 cases, the weak laser field serves similar distribution, which is uniform in p~ in the moving rest ? to the quasiconstant external field example above, giving frame, is shifted in one of the two transverse components, the strong laser pulse a mass density and the ability to remains centered around zero in the other transverse mo- decay at relatively high rapidityy S. mentum component, and remains negligible in direction To create pairs with initially high rapidity in the lab, we parallel to the electrical field. focus on � < �=2. The number of pairs obtained would be For a qualitative model, we take square-wave-localized maximized for the head-on collision� �, but in this case pulses. Setting the waves to be in phase, the oscillation the pair-producing field is at rest in¼ the lab and the pairs within each individual pulse results in a checker-board of appear with zero net momentum. Decreasing� results in stable (E2 B 2 <0) and unstable (E 2 B 2 >0) subdo- � � higher rapidities, but in order to maintain the same yield in mains in the volume where the pulses overlap. The stable particles, intensity of the field in the collision domain must subdomains do not contribute to the particle output. Within be increased, whether by increasing the intensity of the the unstable domains the individual sinusoidal variations s strong pulsea 0 or the ratior. In the limit� 0, the mass of the laser fields produce a variation in the direction and density (and hence the potential for pair production)! van- ~ magnitude of Stot. However, most particles are created near ishes because the converging laser pulses have degenerated the peak of the sinusoid where variation of S~tot is slow. into collinearly propagating waves. Space and time dependence of the fields affects particle As for the above example of a laser field incident on an yields by a factor of order unity and spectra by small external magnetic field, the trade-off between yield of changes to distribution widths. In the following, we evalu- produced pairs and rapidity of the moving rest frame is a ate using the peak field magnitudes and geometry. consequence of the finite total energy density of the laser In unstable subdomains, the magnetic field is reduced by pulses. This trade-off is visible in the� dependence of the cos� in magnitude compared to the net electric field and invariant mass density, which derives from nonvanishing remains orthogonal to it. As a result, d2 in the region of superposition,

033003-3 114 LANCE LABUN AND JOHANN RAFELSKI PHYSICAL REVIEW D 84, 033003 (2011) N 1 dN dy The spatial size of the produced particle bunch is deter- 4 mined by the size of collision volume and, hence, for lasers = /4 focused near to the diffraction limit is � 3 1�m 3. 3.5 = 5 /16 Note that charge separation due to the definite� ¼ orientation ð Þ 3 = 3 /8 of the electric field implies that the positrons appear sepa- 2.5 rated from the electrons by about the width of the pulse. = /2 As N is an extensive quantity, we compute reference 2 valuesh i for a collision volume duration of 5� 3 � ð Þ � 1.5 25�=c. Figure 2 exhibits the trade-off between y; N (right scale, lines with arrows), which arises from theh trade-offi 1 betweeny S; � noted above. 0.5 V. CONCLUSIONS pT m 1 2 3 4 5 6 7 yp The field rest frame is general to classical electromag- FIG. 3 (color online). The normalized distributions in rapidity netism of Maxwell’s equations and applies to most super- and transverse momentum fora s !s=m 1,r 10 2, and� positions of laser pulses. We showed how the invariant 0 ¼ ¼ � ¼ �=2;3�=8;5�=16, and �=4. To the leftp T=m spectra and mass density and the rapidity of the moving rest frame to the right the rapidity spectra. See text for specific meaning are controlled. For the example of two colliding high- ofp T. intensity square laser pulses, we showed how to create collimated high energy electron-positron bunches that are directed away from both laser beam lines and may be �2 2" 2r2 1 cos� 2 TK=4 2: (13) ¼ ð � Þ þ ð K Þ directed independently after creation. The resulting bunches are planar in the plane normal to the electric field The rapidity of the moving rest frame of the field is and Gaussian in momentum. 1 1 2 2 2 We have discussed how the total energy of the unstable 2 2 r� r csc �=2 cot �=2 1 sinh yS ½ð ð þ Þ ð Þþ ð Þ� � : (14) field configuration can be smoothly allotted between the ¼ 1 T K=4"d2 2 þð K Þ energy per particle and total number of particles in the For the strong fields necessary to induce significant pair bunch. In the present model, the particle number in the K resultant bunch depends most on the absolute magnitude of creation,T K =4 is a noticeable correction, comparable in magnitude to focusing corrections. In generating Fig. 2, we the fields attained in the collision volume. Techniques such K as optimizing pulse shape [19] and frequency structure have used nonperturbatively computed values of" andT K [24,26] can be applied to enhance the yield. [8]. Ford&0:1 E~c their series expansions, given in [8], can be used toj goodj accuracy. Particles are produced in their rest frame at a very low In the laboratory frame, the vacuum-derived pairs have a energy, and the high laboratory energy is achieved by superposing nearly collinear, pulsed electromagnetic fields 2 2 boosted mean momentum p S m p T sinh y y S so as to have the rest frame moving at high rapidity relative h k i ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ ð � Þ in the direction of S~tot, thus defining direction of the bunch. to the lab. No acceleration process is needed to bring the charged particles to high energy. In our proposed beam In the plane normal to S~tot, the produced particles have a momentum distribution concentrated in the direction that is geometry, the laser fields may be made very strong without perpendicular also to the net electric field vector. In this endangering the stability of the experiment or its apparatus by near-critical laser pulses. The vacuum-decay products direction, thep T momentum distribution is Gaussian with mean zero, unaffected by the Lorentz transformation along in laser pulse collisions are readily distinguishable from particle-creation backgrounds such as cascades [38] and S~ . With high rapidity and low transverse momenta, the tot perturbative processes thus enhancing discovery potential. vacuum-decay products form a pancaked, tightly colli- The concept we have described may permit future develop- mated bunch with low transverse emittance. Distributions ment of a source of particles with ultrahigh energy. in rapidityy y y and transverse momentump are p ¼ � S T shown in Fig. 3 for four selected collision angles� ACKNOWLEDGMENTS 8�=16,6�=16,5�=16,4�=16. To the left, we see the¼ transverse momentum distribution. To the right, we see This work was supported by a grant from the U.S. normalized distribution iny p, integrated overp T. Department of Energy, DE-FG02-04ER41318.

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