DOCSLIB.ORG

A Non-Perturbative Approach to the Scalar Casimir Effect with Lorentz Symmetry Violation ∗ C.A

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Non-Perturbative Approach to the Scalar Casimir Effect with Lorentz Symmetry Violation ∗ C.A Physics Letters B 807 (2020) 135567 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb A non-perturbative approach to the scalar Casimir effect with Lorentz symmetry violation ∗ C.A. Escobar a, , A. Martín-Ruiz b, O.J. Franca b, Marcos A. G. Garcia c a Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, Ciudad de México 01000, Mexico b Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543, 04510 Ciudad de México, Mexico c Instituto de Física Teórica (IFT), UAM-CSIC, Campus de Cantoblanco, 28049 Madrid, Spain a r t i c l e i n f o a b s t r a c t Article history: We determine the effect of Lorentz invariance violation in the vacuum energy and stress between two Received 30 April 2020 parallel plates separated by a distance L, in the presence of a massive real scalar field. We parametrize the Received in revised form 23 May 2020 Lorentz-violation in terms of a symmetric tensor h μν that represents a constant background. Through the Accepted 16 June 2020 Green’s function method, we obtain the global Casimir energy, the Casimir force between the plates and Available online 20 June 2020 the energy density in a closed analytical form without resorting to perturbative methods. With regards Editor: A. Volovichis √ ˜ μν to the pressure, we find that Fc (L) = F0(L)/ −det h , where F0 is the Lorentz-invariant √expression, ˜ μν ˜ nn Keywords: and L is the plate separation rescaled by the component of h normal to the plates, L = L/ −h . We Casimir effect also analyze the Casimir stress including finite-temperature corrections. The local behavior of the Casimir Thermal Casimir effect energy density is also discussed. Lorentz violation © 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license 3 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . 1. Introduction ical systems [13]. Among theoretical extensions one can list its generalization to spacetimes with non-trivial topologies [14,15], The existence of a zero-point vacuum energy is one of the dynamical boundary conditions [16], and non-Euclidean space- main tenets of the quantum formulation of the laws that we be- times [17–19]. In the latter case it offers an independent derivation lieve govern our Universe. In a Quantum Field Theory (QFT), the of the Hawking temperature from particle production from black presence of fluctuating zero-point fields implies the existence of a holes [20,21]. Modifications of the Casimir effect in the presence non-vanishing macroscopic force between the boundaries that de- of weak gravitational fields have been extensively studied [22–28]. limit a spatial region [1], due to the difference in the spectrum In this context, the Casimir effect can potentially provide clues on of quantized field modes inside and outside this region. When the connection between zero-point fluctuations and the cosmolog- the boundaries of this delimited spatial domain take the form of ical constant [29–32]. two parallel plates, this manifestation of the vacuum fluctuation is The Casimir effect stands as a potential handle to distinguish known as the Casimir effect [2]. The computation of the Casimir between Lorentz-invariant and Lorentz-violating formulations of force in QFT is a standard textbook exercise [3–5], and its exis- QFT. Lorentz invariance (LI) is one of the cornerstones behind QFT tence, in the case of Quantum Electrodynamics, has been verified and general relativity, and to date there are no experimental signs to a high precision [6–9]. of a departure from it [33]. Nevertheless, the quantum nature of The Casimir effect is now behind many experimental and theo- the spacetime at distances of the order of the Planck length (P ) retical pursuits. It is used as a tool to place constraints on Yukawa- has been shown to provide mechanisms that can lead to viola- type interactions [10,11], and it has been suggested as a poten- tion of LI in certain formulations of quantum gravity [34–36]. As tial probe for the detection of feebly-interacting axion-like dark an example, spontaneous Lorentz symmetry breaking can occur matter [12]. Casimir forces cannot be neglected at the nanoscale, within some string theories [35]. Therefore, a better understand- and must be accounted for in the design of microelectromechan- ing on the consequences of the breakdown of LI at scales larger than P would provide valuable information about the microscopic structure of spacetime. In this Letter we explore the manifesta- Corresponding author. * tion of the spontaneous breakdown of LI, induced by a constant E-mail addresses: carlos_escobar@fisica.unam.mx (C.A. Escobar), background tensor, on the Casimir effect for a real massive scalar [email protected] (A. Martín-Ruiz), [email protected] (O.J. Franca), [email protected] field between two parallel conductive plates in flat spacetime. We (M.A.G.Garcia). also explore how thermal corrections are affected by the Lorentz https://doi.org/10.1016/j.physletb.2020.135567 0370-2693/© 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 2 C.A. Escobar et al. / Physics Letters B 807 (2020) 135567 symmetry breaking. Our work provides a generalization of previ- Here z> (z<) is the greater (lesser) between z and z . The coeffi- ous studies of LI violation in the Casimir context [37,38]. cients ξ0,1 denote the following combinations of energy-momenta and the Lorentz-violating tensor, 2. The model 1 03 ¯i3 ξ0 = (h ω − h k⊥¯), (6) Arguably, the most straightforward way to implement Lorentz h33 i violation is by means of the introduction of a tensor field with a 1/2 1 03 ¯i3 2 33 2 ξ1 = (h ω − h k⊥¯) − h γ , (7) non-zero vacuum expectation value (VEV). When coupled to the |h33| i Standard Model fields the spontaneous symmetry breaking, in- duced by the non-zero VEV, is manifested as preferential directions where on the spacetime, leading to a breakdown of LI. In the case of a 2 00 2 0¯i ¯i¯j 2 real scalar field in flat spacetime, with the Minkowski metric with γ = h ω − 2h ωk⊥ + h k⊥ k⊥ − m . (8) ¯i ¯i ¯j signature (+, −, −, −), we parametrize this coupling in the follow- μν → μν ing form, The LI limit is recovered by taking h η , which implies → 2 → 2 − 2 − 2 ξ0 0 and ξ1 ω k⊥ m . It is worth noting that the case 1 μν 1 2 2 with Neumann conditions can be trivially recovered by replacing L = h ∂μφ∂νφ − m φ . (1) 2 2 sin → cos in the numerator of (5). μν The determination of the Casimir energy and stress requires not Here h is a symmetric tensor that represents a constant back- only the GF for the two plate setup, but also the GF in the absence ground, independent of the spacetime position, and which does of plates and a single plate. For the former, we find not transform as a second order tensor under active Lorentz trans- 1 formations. Naturally, causality, the positive energy condition and iξ (z−z ) stability impose restrictions on the components of h μν . i e 0 − g (z, z ) =− eiξ1(z> z<) , (9) v 33 2ξ1 h Consider now the following set-up: a pair of parallel, conduc- tive plates, orthogonal to the zˆ-direction, located at z = 0 and while for the latter, z = L, on which Dirichlet boundary conditions apply for the field φ. − = = = = iξ0(z z ) That is, φ(z 0) φ(z L) 0. We now solve for the scalar field e iξ (z −L) g|(z, z ) =− sin[ξ (z − L)]e 1 > . (10) 33 1 < between the plates, applying the Green’s function technique [40]. ξ1h Namely, we are interested in computing the time-ordered, vac- In order to quantify the Casimir effect, we need an expression uum two-point correlation function G(x, x ) =−i0|T φ(x)φ(x )|0, which as is well known (see e.g. [41]) satisfies the Green’s function for the vacuum expectation value for the stress-energy tensor of μν μα ν μν the scalar field, T = h ∂ φ∂ φ − η L. In terms of the GF, it (GF) equation α can be generically computed as [40] (4) O G(x, x ) = δ (x − x ). (2) x T μν=−i lim hμα ν G x x − μνL (11) ∂α ∂ ( , ) η , Here, in the configuration space, the modified Klein-Gordon oper- x→x ator has the following explicit form, while the VEV of the Lagrangian density can be written as L = 1 μν 2 ¯ ¯¯ ¯ −i lim → h ∂ ∂ − m G(x, x ). O = 00 2 + 0i + i j + 03 + i3 x x 2 μ ν x h ∂0 2h ∂0∂¯i h ∂¯i∂ ¯j 2h ∂0∂3 2h ∂¯i∂3 Substitution of (4)leads to the following expressions for the + h33∂2 + m2 (3) z energy density and the pressure in the zˆ-direction, ¯ ¯ with i, j = 1, 2. In the chosen coordinate system, the GF is invari- 2 ant under translations in the (xˆ, yˆ )-plane. Taking advantage of this 00 dω d k⊥ 00 2 0¯i T =−i lim h ω − h ωk⊥¯ symmetry, we can express the GF in terms of the Fourier transform z→z 2π (2π)2 i in the direction parallel to the plates, 03 +ih ω∂z g(z, z ) −L, (12) 2 d k⊥ ik⊥·(x⊥−x ) dω −iω(t−t ) 2 G(x, x ) = e ⊥ e g z, z ; ,k⊥ , i dω d k⊥ 2 ω 33 2 33 (2π) 2π T =− lim γ − h ∂z∂z g(z, z ).
Load more

Recommended publications
  • An Introduction to Quantum Field Theory
    AN INTRODUCTION TO QUANTUM FIELD THEORY By Dr M Dasgupta University of Manchester Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 1 - - 2 - Contents 0 Prologue....................................................................................................... 5 1 Introduction ................................................................................................ 6 1.1 Lagrangian formalism in classical mechanics......................................... 6 1.2 Quantum mechanics................................................................................... 8 1.3 The Schrödinger picture........................................................................... 10 1.4 The Heisenberg picture............................................................................ 11 1.5 The quantum mechanical harmonic oscillator ..................................... 12 Problems .............................................................................................................. 13 2 Classical Field Theory............................................................................. 14 2.1 From N-point mechanics to field theory ............................................... 14 2.2 Relativistic field theory ............................................................................ 15 2.3 Action for a scalar field ............................................................................ 15 2.4 Plane wave solution to the Klein-Gordon equation ...........................
    [Show full text]
  • Physical Masses and the Vacuum Expectation Value
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE PHYSICAL MASSES AND THE VACUUM EXPECTATION VALUE provided by CERN Document Server OF THE HIGGS FIELD Hung Cheng Department of Mathematics, Massachusetts Institute of Technology Cambridge, MA 02139, U.S.A. and S.P.Li Institute of Physics, Academia Sinica Nankang, Taip ei, Taiwan, Republic of China Abstract By using the Ward-Takahashi identities in the Landau gauge, we derive exact relations between particle masses and the vacuum exp ectation value of the Higgs eld in the Ab elian gauge eld theory with a Higgs meson. PACS numb ers : 03.70.+k; 11.15-q 1 Despite the pioneering work of 't Ho oft and Veltman on the renormalizability of gauge eld theories with a sp ontaneously broken vacuum symmetry,a numb er of questions remain 2 op en . In particular, the numb er of renormalized parameters in these theories exceeds that of bare parameters. For such theories to b e renormalizable, there must exist relationships among the renormalized parameters involving no in nities. As an example, the bare mass M of the gauge eld in the Ab elian-Higgs theory is related 0 to the bare vacuum exp ectation value v of the Higgs eld by 0 M = g v ; (1) 0 0 0 where g is the bare gauge coupling constant in the theory.We shall show that 0 v u u D (0) T t g (0)v: (2) M = 2 D (M ) T 2 In (2), g (0) is equal to g (k )atk = 0, with the running renormalized gauge coupling constant de ned in (27), and v is the renormalized vacuum exp ectation value of the Higgs eld de ned in (29) b elow.
    [Show full text]
  • B1. the Generating Functional Z[J]
    B1. The generating functional Z[J] The generating functional Z[J] is a key object in quantum field theory - as we shall see it reduces in ordinary quantum mechanics to a limiting form of the 1-particle propagator G(x; x0jJ) when the particle is under thje influence of an external field J(t). However, before beginning, it is useful to look at another closely related object, viz., the generating functional Z¯(J) in ordinary probability theory. Recall that for a simple random variable φ, we can assign a probability distribution P (φ), so that the expectation value of any variable A(φ) that depends on φ is given by Z hAi = dφP (φ)A(φ); (1) where we have assumed the normalization condition Z dφP (φ) = 1: (2) Now let us consider the generating functional Z¯(J) defined by Z Z¯(J) = dφP (φ)eφ: (3) From this definition, it immediately follows that the "n-th moment" of the probability dis- tribution P (φ) is Z @nZ¯(J) g = hφni = dφP (φ)φn = j ; (4) n @J n J=0 and that we can expand Z¯(J) as 1 X J n Z Z¯(J) = dφP (φ)φn n! n=0 1 = g J n; (5) n! n ¯ so that Z(J) acts as a "generator" for the infinite sequence of moments gn of the probability distribution P (φ). For future reference it is also useful to recall how one may also expand the logarithm of Z¯(J); we write, Z¯(J) = exp[W¯ (J)] Z W¯ (J) = ln Z¯(J) = ln dφP (φ)eφ: (6) 1 Now suppose we make a power series expansion of W¯ (J), i.e., 1 X 1 W¯ (J) = C J n; (7) n! n n=0 where the Cn, known as "cumulants", are just @nW¯ (J) C = j : (8) n @J n J=0 The relationship between the cumulants Cn and moments gn is easily found.
    [Show full text]
  • Finite Quantum Field Theory and Renormalization Group
    Finite Quantum Field Theory and Renormalization Group M. A. Greena and J. W. Moffata;b aPerimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada bDepartment of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada September 15, 2021 Abstract Renormalization group methods are applied to a scalar field within a finite, nonlocal quantum field theory formulated perturbatively in Euclidean momentum space. It is demonstrated that the triviality problem in scalar field theory, the Higgs boson mass hierarchy problem and the stability of the vacuum do not arise as issues in the theory. The scalar Higgs field has no Landau pole. 1 Introduction An alternative version of the Standard Model (SM), constructed using an ultraviolet finite quantum field theory with nonlocal field operators, was investigated in previous work [1]. In place of Dirac delta-functions, δ(x), the theory uses distributions (x) based on finite width Gaussians. The Poincar´eand gauge invariant model adapts perturbative quantumE field theory (QFT), with a finite renormalization, to yield finite quantum loops. For the weak interactions, SU(2) U(1) is treated as an ab initio broken symmetry group with non- zero masses for the W and Z intermediate× vector bosons and for left and right quarks and leptons. The model guarantees the stability of the vacuum. Two energy scales, ΛM and ΛH , were introduced; the rate of asymptotic vanishing of all coupling strengths at vertices not involving the Higgs boson is controlled by ΛM , while ΛH controls the vanishing of couplings to the Higgs. Experimental tests of the model, using future linear or circular colliders, were proposed.
    [Show full text]
  • U(1) Symmetry of the Complex Scalar and Scalar Electrodynamics
    Fall 2019: Classical Field Theory (PH6297) U(1) Symmetry of the complex scalar and scalar electrodynamics August 27, 2019 1 Global U(1) symmetry of the complex field theory & associated Noether charge Consider the complex scalar field theory, defined by the action, h i h i I Φ(x); Φy(x) = d4x (@ Φ)y @µΦ − V ΦyΦ : (1) ˆ µ As we have noted earlier complex scalar field theory action Eq. (1) is invariant under multiplication by a constant complex phase factor ei α, Φ ! Φ0 = e−i αΦ; Φy ! Φ0y = ei αΦy: (2) The phase,α is necessarily a real number. Since a complex phase is unitary 1 × 1 matrix i.e. the complex conjugation is also the inverse, y −1 e−i α = e−i α ; such phases are also called U(1) factors (U stands for Unitary matrix and since a number is a 1×1 matrix, U(1) is unitary matrix of size 1 × 1). Since this symmetry transformation does not touch spacetime but only changes the fields, such a symmetry is called an internal symmetry. Also note that since α is a constant i.e. not a function of spacetime, it is a global symmetry (global = same everywhere = independent of spacetime location). Check: Under the U(1) symmetry Eq. (2), the combination ΦyΦ is obviously invariant, 0 Φ0yΦ = ei αΦy e−i αΦ = ΦyΦ: This implies any function of the product ΦyΦ is also invariant. 0 V Φ0yΦ = V ΦyΦ : Note that this is true whether α is a constant or a function of spacetime i.e.
    [Show full text]
  • Vacuum Energy
    Vacuum Energy Mark D. Roberts, 117 Queen’s Road, Wimbledon, London SW19 8NS, Email:[email protected] http://cosmology.mth.uct.ac.za/ roberts ∼ February 1, 2008 Eprint: hep-th/0012062 Comments: A comprehensive review of Vacuum Energy, which is an extended version of a poster presented at L¨uderitz (2000). This is not a review of the cosmolog- ical constant per se, but rather vacuum energy in general, my approach to the cosmological constant is not standard. Lots of very small changes and several additions for the second and third versions: constructive feedback still welcome, but the next version will be sometime in coming due to my sporadiac internet access. First Version 153 pages, 368 references. Second Version 161 pages, 399 references. arXiv:hep-th/0012062v3 22 Jul 2001 Third Version 167 pages, 412 references. The 1999 PACS Physics and Astronomy Classification Scheme: http://publish.aps.org/eprint/gateway/pacslist 11.10.+x, 04.62.+v, 98.80.-k, 03.70.+k; The 2000 Mathematical Classification Scheme: http://www.ams.org/msc 81T20, 83E99, 81Q99, 83F05. 3 KEYPHRASES: Vacuum Energy, Inertial Mass, Principle of Equivalence. 1 Abstract There appears to be three, perhaps related, ways of approaching the nature of vacuum energy. The first is to say that it is just the lowest energy state of a given, usually quantum, system. The second is to equate vacuum energy with the Casimir energy. The third is to note that an energy difference from a complete vacuum might have some long range effect, typically this energy difference is interpreted as the cosmological constant.
    [Show full text]
  • 8. Quantum Field Theory on the Plane
    8. Quantum Field Theory on the Plane In this section, we step up a dimension. We will discuss quantum field theories in d =2+1dimensions.Liketheird =1+1dimensionalcounterparts,thesetheories have application in various condensed matter systems. However, they also give us further insight into the kinds of phases that can arise in quantum field theory. 8.1 Electromagnetism in Three Dimensions We start with Maxwell theory in d =2=1.ThegaugefieldisAµ,withµ =0, 1, 2. The corresponding field strength describes a single magnetic field B = F12,andtwo electric fields Ei = F0i.Weworkwiththeusualaction, 1 S = d3x F F µ⌫ + A jµ (8.1) Maxwell − 4e2 µ⌫ µ Z The gauge coupling has dimension [e2]=1.Thisisimportant.ItmeansthatU(1) gauge theories in d =2+1dimensionscoupledtomatterarestronglycoupledinthe infra-red. In this regard, these theories di↵er from electromagnetism in d =3+1. We can start by thinking classically. The Maxwell equations are 1 @ F µ⌫ = j⌫ e2 µ Suppose that we put a test charge Q at the origin. The Maxwell equations reduce to 2A = Qδ2(x) r 0 which has the solution Q r A = log +constant 0 2⇡ r ✓ 0 ◆ for some arbitrary r0. We learn that the potential energy V (r)betweentwocharges, Q and Q,separatedbyadistancer, increases logarithmically − Q2 r V (r)= log +constant (8.2) 2⇡ r ✓ 0 ◆ This is a form of confinement, but it’s an extremely mild form of confinement as the log function grows very slowly. For obvious reasons, it’s usually referred to as log confinement. –384– In the absence of matter, we can look for propagating degrees of freedom of the gauge field itself.
    [Show full text]
  • Physics 234C Lecture Notes
    Physics 234C Lecture Notes Jordan Smolinsky [email protected] Department of Physics & Astronomy, University of California, Irvine, ca 92697 Abstract These are lecture notes for Physics 234C: Advanced Elementary Particle Physics as taught by Tim M.P. Tait during the spring quarter of 2015. This is a work in progress, I will try to update it as frequently as possible. Corrections or comments are always welcome at the above email address. 1 Goldstone Bosons Goldstone's Theorem states that there is a massless scalar field for each spontaneously broken generator of a global symmetry. Rather than prove this, we will take it as an axiom but provide a supporting example. Take the Lagrangian given below: N X 1 1 λ 2 L = (@ φ )(@µφ ) + µ2φ φ − (φ φ ) (1) 2 µ i i 2 i i 4 i i i=1 This is a theory of N real scalar fields, each of which interacts with itself by a φ4 coupling. Note that the mass term for each of these fields is tachyonic, it comes with an additional − sign as compared to the usual real scalar field theory we know and love. This is not the most general such theory we could have: we can imagine instead a theory which includes mixing between the φi or has a nondegenerate mass spectrum. These can be accommodated by making the more general replacements X 2 X µ φiφi ! Mijφiφj i i;j (2) X 2 X λ (φiφi) ! Λijklφiφjφkφl i i;j;k;l but this restriction on the potential terms of the Lagrangian endows the theory with a rich structure.
    [Show full text]
  • Physics 215A: Particles and Fields Fall 2016
    Physics 215A: Particles and Fields Fall 2016 Lecturer: McGreevy These lecture notes live here. Please email corrections to mcgreevy at physics dot ucsd dot edu. Last updated: 2017/10/31, 16:42:12 1 Contents 0.1 Introductory remarks............................4 0.2 Conventions.................................6 1 From particles to fields to particles again7 1.1 Quantum sound: Phonons.........................7 1.2 Scalar field theory.............................. 14 1.3 Quantum light: Photons.......................... 19 1.4 Lagrangian field theory........................... 24 1.5 Casimir effect: vacuum energy is real................... 29 1.6 Lessons................................... 33 2 Lorentz invariance and causality 34 2.1 Causality and antiparticles......................... 36 2.2 Propagators, Green's functions, contour integrals............ 40 2.3 Interlude: where is 1-particle quantum mechanics?............ 45 3 Interactions, not too strong 48 3.1 Where does the time dependence go?................... 48 3.2 S-matrix................................... 52 3.3 Time-ordered equals normal-ordered plus contractions.......... 54 3.4 Time-ordered correlation functions.................... 57 3.5 Interlude: old-fashioned perturbation theory............... 67 3.6 From correlation functions to the S matrix................ 69 3.7 From the S-matrix to observable physics................. 78 4 Spinor fields and fermions 85 4.1 More on symmetries in QFT........................ 85 4.2 Representations of the Lorentz group on fields.............. 91 4.3 Spinor lagrangians............................. 96 4.4 Free particle solutions of spinor wave equations............. 103 2 4.5 Quantum spinor fields........................... 106 5 Quantum electrodynamics 111 5.1 Vector fields, quickly............................ 111 5.2 Feynman rules for QED.......................... 115 5.3 QED processes at leading order...................... 120 3 0.1 Introductory remarks Quantum field theory (QFT) is the quantum mechanics of extensive degrees of freedom.
    [Show full text]
  • Quantum Mechanics Propagator
    Quantum Mechanics_propagator This article is about Quantum field theory. For plant propagation, see Plant propagation. In Quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. In Feynman diagrams, which calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the diagram. They also can be viewed as the inverse of the wave operator appropriate to the particle, and are therefore often called Green's functions. Non-relativistic propagators In non-relativistic quantum mechanics the propagator gives the probability amplitude for a particle to travel from one spatial point at one time to another spatial point at a later time. It is the Green's function (fundamental solution) for the Schrödinger equation. This means that, if a system has Hamiltonian H, then the appropriate propagator is a function satisfying where Hx denotes the Hamiltonian written in terms of the x coordinates, δ(x)denotes the Dirac delta-function, Θ(x) is the Heaviside step function and K(x,t;x',t')is the kernel of the differential operator in question, often referred to as the propagator instead of G in this context, and henceforth in this article. This propagator can also be written as where Û(t,t' ) is the unitary time-evolution operator for the system taking states at time t to states at time t'. The quantum mechanical propagator may also be found by using a path integral, where the boundary conditions of the path integral include q(t)=x, q(t')=x' .
    [Show full text]
  • The Quantum Vacuum and the Cosmological Constant Problem
    The Quantum Vacuum and the Cosmological Constant Problem S.E. Rugh∗and H. Zinkernagel† Abstract - The cosmological constant problem arises at the intersection be- tween general relativity and quantum field theory, and is regarded as a fun- damental problem in modern physics. In this paper we describe the historical and conceptual origin of the cosmological constant problem which is intimately connected to the vacuum concept in quantum field theory. We critically dis- cuss how the problem rests on the notion of physical real vacuum energy, and which relations between general relativity and quantum field theory are assumed in order to make the problem well-defined. Introduction Is empty space really empty? In the quantum field theories (QFT’s) which underlie modern particle physics, the notion of empty space has been replaced with that of a vacuum state, defined to be the ground (lowest energy density) state of a collection of quantum fields. A peculiar and truly quantum mechanical feature of the quantum fields is that they exhibit zero-point fluctuations everywhere in space, even in regions which are otherwise ‘empty’ (i.e. devoid of matter and radiation). These zero-point arXiv:hep-th/0012253v1 28 Dec 2000 fluctuations of the quantum fields, as well as other ‘vacuum phenomena’ of quantum field theory, give rise to an enormous vacuum energy density ρvac. As we shall see, this vacuum energy density is believed to act as a contribution to the cosmological constant Λ appearing in Einstein’s field equations from 1917, 1 8πG R g R Λg = T (1) µν − 2 µν − µν c4 µν where Rµν and R refer to the curvature of spacetime, gµν is the metric, Tµν the energy-momentum tensor, G the gravitational constant, and c the speed of light.
    [Show full text]
  • Path Integral in Quantum Field Theory Alexander Belyaev (Course Based on Lectures by Steven King) Contents
    Path Integral in Quantum Field Theory Alexander Belyaev (course based on Lectures by Steven King) Contents 1 Preliminaries 5 1.1 Review of Classical Mechanics of Finite System . 5 1.2 Review of Non-Relativistic Quantum Mechanics . 7 1.3 Relativistic Quantum Mechanics . 14 1.3.1 Relativistic Conventions and Notation . 14 1.3.2 TheKlein-GordonEquation . 15 1.4 ProblemsSet1 ........................... 18 2 The Klein-Gordon Field 19 2.1 Introduction............................. 19 2.2 ClassicalScalarFieldTheory . 20 2.3 QuantumScalarFieldTheory . 28 2.4 ProblemsSet2 ........................... 35 3 Interacting Klein-Gordon Fields 37 3.1 Introduction............................. 37 3.2 PerturbationandScatteringTheory. 37 3.3 TheInteractionHamiltonian. 43 3.4 Example: K π+π− ....................... 45 S → 3.5 Wick’s Theorem, Feynman Propagator, Feynman Diagrams . .. 47 3.6 TheLSZReductionFormula. 52 3.7 ProblemsSet3 ........................... 58 4 Transition Rates and Cross-Sections 61 4.1 TransitionRates .......................... 61 4.2 TheNumberofFinalStates . 63 4.3 Lorentz Invariant Phase Space (LIPS) . 63 4.4 CrossSections............................ 64 4.5 Two-bodyScattering . 65 4.6 DecayRates............................. 66 4.7 OpticalTheorem .......................... 66 4.8 ProblemsSet4 ........................... 68 1 2 CONTENTS 5 Path Integrals in Quantum Mechanics 69 5.1 Introduction............................. 69 5.2 The Point to Point Transition Amplitude . 70 5.3 ImaginaryTime........................... 74 5.4 Transition Amplitudes With an External Driving Force . ... 77 5.5 Expectation Values of Heisenberg Position Operators . .... 81 5.6 Appendix .............................. 83 5.6.1 GaussianIntegration . 83 5.6.2 Functionals ......................... 85 5.7 ProblemsSet5 ........................... 87 6 Path Integral Quantisation of the Klein-Gordon Field 89 6.1 Introduction............................. 89 6.2 TheFeynmanPropagator(again) . 91 6.3 Green’s Functions in Free Field Theory .
    [Show full text]