Hindawi Publishing Corporation Advances in Mathematical Volume 2015, Article ID 197197, 8 pages http://dx.doi.org/10.1155/2015/197197

Research Article Thermal Ground State and Nonthermal Probes

Thierry Grandou1 and Ralf Hofmann2

1 Institut Non Lineaire´ de Nice, 1361 route des Lucioles, Sophia Antipolis, 06560 Valbonne, France 2Institut fur¨ Theoretische Physik, Universitat¨ Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

Correspondence should be addressed to Ralf Hofmann; [email protected]

Received 31 August 2015; Accepted 27 October 2015

Academic Editor: Ivan Avramidi

Copyright © 2015 T. Grandou and R. Hofmann. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The Euclidean formulation of SU(2) Yang-Mills thermodynamics admits periodic, (anti)self-dual solutions to the fundamental, classical equation of motion which possess one unit of topological charge: (anti)calorons. A spatial coarse graining over the central region in a pair of such localised field configurations with trivial holonomy generates an inert adjoint scalar field 𝜙,effectively describing the pure quantum part of the thermal ground state in the induced . Here we show for the limit of zero holonomy how (anti)calorons associate a temperature independent electric permittivity and magnetic permeability to the thermal (2) ground state of SU CMB, the Yang-Mills theory conjectured to underlie the fundamental description of thermal photon gases.

1. Introduction and Minkowskian time evolution as expressed by a time- periodic SU(2) (anti)self-dual gauge field configuration—a is a highly efficient framework to (anti)caloron—whose ℏ is associated with one unit describe the subatomic world [1–3], including coherence of winding about a central spacetime point gives rise to phenomena that extend to macroscopic length and time indeterminism in the process it mediates. That each unit −1 scales [4–10]. The key quantity to describe deviations from of action assigned to (anti)calorons of radius 𝜌=|𝜙|, classicalbehaviorisPlanck’squantumofactionℏ=ℎ/2𝜋= 16 which dominate the emergence of the thermal ground state, 6.58 × 10 eV s which determines the fundamental interac- equals ℏ follows from the value of the coupling 𝑒 in the tion between charged matter and the electromagnetic field induced, effective, thermal quantum field theory [13–17] of andthusalsotheshapeofblackbodyspectrabyrelating the deconfining phase in SU(2) Yang-Mills thermodynamics. frequency 𝜔 andwavevectork to particle-like energy 𝐸=ℏ𝜔 The coupling 𝑒,inturn,obeysanevolutionintemperature and momentum p =ℏk and by appeal to Bose-Einstein (flat almost everywhere) which represents the validity of statistics. In Quantum Mechanics, ℏ sets the strength of Legendre transformations in the effective ensemble where the multiplicative noncommutativity for a pair of canonically thermal ground state coexists with massive (adjoint Higgs conjugate variables such as position and momentum, imply- mechanism) and massless (intact U(1))thermalfluctuations. ing the respective uncertainty relations. The thermal ground state thus is a spatially homogeneous Despite being generally accepted as a universal constant ensemble of quantum fluctuations carried by (anti)caloron of nature and in spite of the fact that we are able to efficiently centers.Atthesametime,aswewillsee,thisstateprovides compute quantum mechanical amplitudes and quantum electric and magnetic dipole densities supporting the prop- statistical averages for a vast variety of processes in parti- agation of certain electromagnetic waves in an SU(2) Yang- Λ∼10−4 (2) cle collisions, atoms, and molecules, extended condensed- Mills of scale eV, SU CMB [18]. matter systems, and astrophysical objects to match exper- In the present work, we establish this link between iment and observation very well, one should remain curi- quantised action, represented by 𝜙,andclassicalwaveprop- ous concerning the principle mechanism that causes the agationenabledbythevacuumparameters𝜖0 and 𝜇0 in emergence of a universal quantum of action. In [11, 12] it terms of the central and peripheral structure of a trivial- was argued that the irreconcilability of classical Euclidean holonomy (anti)caloron, respectively. That is, by allowing 2 Advances in Mathematical Physics afictitioustemperature𝑇 to represent the energy density of (unitary-Coulomb) characterised by a massless mode (𝛾, an electromagnetic wave (nonthermal, external probe) via unbroken U(1) subgroup of SU(2)) and two thermal quasi- ± the thermal ground state through which it propagates we ask particle modes of equal mass 𝑚 = 2𝑒|𝜙| (𝑉 ,massinduced what this implies for 𝜖0 and 𝜇0.Asaresult,both𝜖0 and 𝜇0 by adjoint Higgs mechanism) which propagate thermally, neither depend on 𝑇 nor, as we will argue, on a singled-out that is, on-shell only. Interactions within this propagating inertial frame provided that a certain condition on intensity sector are mediated by isolated (anti)calorons whose action is and frequency of the wave is satisfied. This is a first step in argued to be ℏ [11, 12]. Judged in terms of inclusive quantities reviving the concept of the luminiferous aether, albeit now such as radiative corrections to the one-loop pressure or with the goal of constructing a Poincareinvariantobject.´ the energy density of blackbody radiation, these interactions This paper is organised as follows. In the next section are feeble, and their expansion into 1-PI irreducible bubble we shorty discuss key features of the effective theory for diagrams is conjectured to terminate at a finite number of ± the deconfining phase of SU(2) Yang-Mills thermodynamics. loops [18]. However, spectrally seen, the effects of 𝑉 inter- Section 3 contains a reminder to principles in interpreting acting with 𝛾 lead to severe consequences at low frequencies a Euclidean field configuration in terms of Minkowskian and temperatures comparable to the critical temperature 𝑇𝑐 observables. In a next step, general facts are reviewed on where screened (anti)monopoles, released by (anti)caloron Euclidean, periodic, (anti)self-dual field configurations of dissociation upon large-holonomy deformations [20], rapidly charge modulus unity concerning the central locus of action, become massless and thus start to condense. their holonomy, and their behaviour under semiclassical deformation. Finally, we review the anatomy of a zero- 3. Caloron Structure holonomy Harrington-Shepard (HS) caloron in detail, point- ing out its staticity for spatial distances from the center 3.1. Euclidean Field Theory and Interpretable Quantities. Non- that exceed the inverse of temperature, and discuss which trivial solutions to an elliptic differential equation, such as static charge configuration it resembles depending on two the Euclidean Yang-Mills equation 𝐷𝜇𝐹𝜇] =0,nolonger distinct spatial distance regimes. In Section 4 we briefly are solutions of the corresponding hyperbolic equation upon review the postulate that an SU(2) Yang-Mills theory of analytic continuation 𝑥4 ≡𝜏 → 𝑖𝑥0 (Wick rotation). To Λ∼10−4 (2) scale eV, SU CMB, describes thermal photon endow meaning to quantities computed on classical field con- gases [18]. Subsequently, the large-distance regime in an HS figurations on a 4D Euclidean spacetime in SU(2) Yang-Mills (anti)caloron is considered in order to deduce an expression thermodynamics in terms of observables in a Minkowskian for 𝜖0 based on knowledge about the electric dipole moment spacetimewethusmustinsistthatthesequantitiesarenot −1 provided by a (anti)caloron of radius radius 𝜌=|𝜙| ,the affected by the Wick rotation. That is, to assign a real-world 𝑉 size of the spatial coarse-graining volume cg, and the fact interpretation to a Euclidean quantity it needs to be (i) that the energy density of the probe must match that of either stationary (not depend on 𝜏) or (ii) associated with the thermal ground state. As a result, 𝜖0 and 𝜇0 turn out an instant in Euclidean spacetime because, by exploiting time to be 𝑇-independent, the former representing an electric translational invariance of the Yang-Mills action, this instant charge, large on the scale of the electron charge, of the canbepickedas(𝜏 = 0, x) in Euclidean spacetime. fictitious constituent monopoles giving rise to the associated dipole density. Zooming in to smaller spatial distances to the 3.2. Review of General Facts. If not stated otherwise we work center, the HS (anti)caloron exhibits isolated (anti)self-dual in supernatural units, ℏ=𝑐=𝑘𝐵 =1,whereℏ is the monopoles. For them to turn into dipoles shaking by the reduced quantum of action, 𝑐 is the speed of light in vacuum, probe fields is required. We then show that the definitions and 𝑘𝐵 is Boltzmann’s constant. A trivial-holonomy caloron 𝜖 𝜇 3 of 0 and 0, which were successfully applied to the large- of topological charge unity on the cylinder 𝑆1 × R ,where distance regime, become meaningless. Finally, our results 𝑆1 isthecircleofcircumference𝛽≡1/𝑇(𝑇 temperature) are discussed. Section 5 summarises the paper and discusses describing the compactified Euclidean time dimension (0≤ 𝜖 𝜇 how a universality of 0 and 0 for the entire, experimentally 𝜏≤𝛽), is constructed by an appropriate superposition investigated electromagnetic spectrum could emerge. of charge-one singular-gauge prepotentials [21, 22] with the temporal coordinate of their instanton centers 2. Sketch of Deconfining SU(2) equidistantly stacked along the infinitely extended Euclidean time dimension [23] to enforce temporal periodicity, 𝐴𝜇(𝜏 = Yang-Mills Thermodynamics 0, x)=𝐴𝜇(𝜏 = 𝛽, x). For gauge group SU(2) this Harrington- Shepard (HS) caloron is given as (antihermitian generators For deconfining SU(2) Yang-Mills thermodynamics, a spatial 𝑡𝑎 (𝑎=1,2,3)with tr 𝑡𝑎𝑡𝑏 = −(1/2)𝛿𝑎𝑏): coarse graining over the (anti)self-dual, that is, the non- 𝑎 propagating [19], topological sector with charge modulus 𝐴𝜇 = 𝜂𝜇]𝑡𝑎𝜕] log Π (𝜏,) 𝑟 , (1) |𝑄| = 1 can be performed; see [18] and references therein, 𝑟≡|x| 𝜂𝑎 to yield an inert adjoint scalar field 𝜙.Itsmodulus|𝜙| sets where , 𝜇] denotes the antiself-dual ’t Hooft symbol, 𝜂𝑎 =𝜖𝑎 −𝛿 𝛿 +𝛿 𝛿 the maximal possible resolution in the effective theory whose 𝜇] 𝜇] 𝑎𝜇 ]4 𝑎] 𝜇4,and 6 2 ground state energy density essentially is given as tr(Λ /𝜙 )= 2 3 𝜋𝜌 sinh (2𝜋𝑟/𝛽) 4𝜋Λ 𝑇 (Λ, a constant of integration of dimension mass) and Π (𝜏,) 𝑟 =1+ . 𝛽𝑟 (2𝜋𝑟/𝛽) − (2𝜋𝜏/𝛽) (2) whose propagating sector is, in a totally fixed, physical gauge cosh cos Advances in Mathematical Physics 3

Here 𝜌 isthescaleparameterofthesingular-gaugeinstanton and to repulsion in the complementary range of (large) 1 3 to seed the “mirror sum” within 𝑆 × R ,leadingto(2). holonomy. Because there is no localised counterpart to a The associated antiself-dual field configuration is obtained in monopole or antimonopole in the trivial-holonomy limit, HS 𝑎 𝑎 replacing 𝜂𝜇] by 𝜂𝜇] (self-dual ’t Hooft symbol) in (1). calorons must be considered stable under Gaussian fluctua- Configuration (1) is singular at 𝜏=𝑟=0.Thispoint tions, in contrast to the case of nontrivial holonomy which is is the locus of the configuration’s topological charge 𝑄=1 unstable. The latter statement is also mirrored by the fact that in the sense that the integral of the Chern-Simons current a nontrivial, static holonomy leads to zero quantum weight in 2 𝑎 𝑎 𝑎𝑏𝑐 𝑎 𝑏 𝑐 the infinite-volume limit (which is realistic at high tempera- 𝐾𝜇 = (1/16𝜋 )𝜖𝜇𝛼𝛽𝛾(𝐴𝛼𝜕𝛽𝐴𝛾 + (1/3)𝜖 𝐴𝛼𝐴𝛽𝐴𝛾) over a 3 tures [18] where the radius of the spatial coarse-graining vol- three-sphere 𝑆 of radius 𝛿, which is centered there, yields −1 𝛿 ume for a single caloron diverges as |𝜙| = √2𝜋𝑇/Λ3,where unity independently of 𝛿≥0. Self-duality implies that the Λ being the Yang-Mills scale). As a consequence, nontrivial action of the HS caloron is given as holonomy can only occur transiently in configurations which 8𝜋2 8𝜋2 do not saturate (anti)self-duality bounds to the Yang-Mills 𝑆 = ∫ 𝑑Σ 𝐾 = , 𝐶 2 𝜇 𝜇 2 (3) action. Again, this is equivalent to stating the instability of the 𝑔 𝑆3 𝑔 𝛿 LLKvB solution. It can be shown [18] that the small-holonomy case of monopole-antimonopole attraction by far dominates where 𝑔 is the coupling constant in Euclidean (classical) the situation of monopole-antimonopole repulsion when a theory. Equation (3) holds in the limit 𝛿→0,meaningthat caloron dissociates into its constituents. 𝑆𝐶 canbeattributedtothesingularityoftheHSsolutionat The spatial coarse graining over (anti)self-dual calorons 𝜏=𝑟=0and thus has a Minkowskian interpretation; see of charge modulus |𝑄| =, 1 which do not propagate (due Section 3.1. Based on [13–16] and on the fact that the thermal to (anti)self-duality their energy-momentum tensor vanishes ground state emerges from |𝑄| = 1 caloron/anticalorons, identically [19]), yielding a highly accurate a priori estimate whose scale parameter 𝜌 essentially coincides with the inverse −1 of the deconfining thermal ground state in terms of an inert, of maximal resolution, |𝜙| , in the effective theory for decon- gs adjoint scalar field 𝜙 and a pure-gauge configuration 𝑎𝜇 ,is fining SU(2) Yang-Mills thermodynamics, it was argued in performed over isolated and stable HS solutions [18]. The [11] (see also [12]) that 𝑆𝐶 (as well as the action of an HS gs −1 coarse-grained field 𝑎𝜇 represents a posteriori the effects of anticaloron 𝑆𝐴 with 𝜌∼|𝜙| )equalsℏ iftheeffectivetheory is to be interpreted as a local quantum field theory. small holonomy changes due to (anti)caloron overlap and The HS caloron is the trivial-holonomy limit of the self- interaction. dual Lee-Lu-Kraan-van-Baal (LLKvB) configuration with 𝑄=1and total magnetic charge zero [24–26] which is 3.3. Anatomy of a Relevant Harrington-Shepard Caloron. Let constructed via the Nahm transformation of self-dual fields usnowreview[30]howthefieldstrengthofanHScaloron on the Euclidean four torus [27–29]. For nontrivial holonomy depends on the distance from its center at 𝜏=𝑟=0.For 3 |𝑥| ≪ 𝛽 |𝑥| ≡ √𝑥2 ≡ 𝑥 𝑥 𝑥 ≡𝜏 (𝐴4(𝑟 → ∞) = 𝑖𝑢𝑡 with 0<𝑢<2𝜋/𝛽) the LLKvB ( √ 𝜇 𝜇, 4 )onehas solution exhibits a pair of a magnetic monopole (m) and itsantimonopole(a)withrespecttotheAbeliansubgroup 𝜋 𝑠 𝜌2 𝑥2 Π (𝑥) = (1+ ) + +𝑂( ) , (6) U(1) ⊂ SU(2) left unbroken by 𝐴4(𝑟 → ∞) =0̸ .Their 3 𝛽 𝑥2 𝛽2 masses are 𝑚𝑚 =4𝜋𝑢and 𝑚𝑎 =4𝜋(2𝜋/𝛽−𝑢)such that, in the trivial-holonomy limits 𝑢 → 0, 2𝜋/𝛽, one of these magnetic where 𝑠 isdefinedin(4).From(6)and(1)oneobtainswith constituents becomes massless and thus completely spatially |𝑥| ≪ 𝛽 the following expression for 𝐹𝜇] = (1/2)𝜖𝜇]𝜅𝜆𝐹𝜅𝜆 ≡ delocalised. For nontrivial holonomy, where both monopole ̃ 𝐹𝜇]: andantimonopoleareoffinitemass,localised,andseparated by a spatial distance 𝜂𝑎 2 𝑎 󸀠2 𝛼𝛽 𝑥 𝐹 =−4𝜌 𝐼𝛼𝜇𝐼𝛽] +𝑂( ) , (7) 𝜌2 𝜇] (𝑥2 +𝜌󸀠2)2 𝛽4 𝑠=𝜋 , 𝛽 (4) 2 where 𝐼𝛼𝜇 ≡𝛿𝛼𝜇 −2(𝑥𝛼𝑥𝜇/𝑥 ). At small four-dimensional dis- they can be considered static by an exact cancellation of tances from the caloron center the field strength thus behaves attraction, mediated by their U(1) magnetic fields, and repul- like the one of a singular-gauge instanton with a renormalised 󸀠2 2 sion due to the field 𝐴4. As was shown in [20] by investigating scale parameter 𝜌 =𝜌/(1+(𝜋/3)(𝑠/𝛽)). Therefore, the field the effective action of a LLKvB caloron (integrating out strength of the HS solution exhibits a dependence on 𝜏 and as Gaussian fluctuations), this balance is distorted, leading to such has no Minkowskian interpretation; see Section 3.1. monopole-antimonopole attraction for For a Minkowskian spacetime one can infer, however, that the action of the configuration is attributable to winding 𝜋 1 0<𝑢≤ (1 − ) of the caloron around the group manifold 𝑆3 as induced by 𝛽 √3 a spacetime point, the instanton center. This is because, in (5) 𝜋 1 2𝜋 the sense of (3), an instant has no analytic continuation or (1 + )<𝑢≤ , Wick rotation. (The 4D action or topological-charge density or 𝛽 √ 𝛽 3 of the caloron is regular at 𝜏=𝑟=0, does depend on 4 Advances in Mathematical Physics

Euclidean spacetime in the vicinity of this point, and thus has the magnetic permeability 𝜇0,canbereducedtothephysics no Minkowskian interpretation.) of the static, non-Abelian, and (anti)self-dual monopole and 𝑎 For 𝑟≫𝛽the self-dual electric and magnetic fields 𝐸𝑖 dipole configurations represented by HS (anti)calorons in the 𝑎 and 𝐵𝑖 are static and can be written as regimes 𝛽≪𝑟≪𝑠and 𝑟≫𝑠≫𝛽,respectively;see Section3.3.Todothis,theconceptofathermalgroundstate 𝑎 2 𝑎 𝑎 (𝑥̂ 𝑥̂𝑖/𝑟 )−(1/𝑟𝑠) (𝛿𝑖 −3𝑥̂ 𝑥̂𝑖) together with information on how it is obtained [18] as well 𝐸𝑎 =𝐵𝑎 ∼− , (8) 𝑖 𝑖 (1 + 𝑟/𝑠)2 as the results of Section 3.3 [30] are invoked.

𝑎 𝑎 where 𝑥̂𝑖 ≡𝑥𝑖/𝑟 and 𝑥̂ ≡𝑥/𝑟.For𝛽≪𝑟≪𝑠(8) simplifies 4.1. Preexisting Dipole Densities. Let us discuss the case as 𝑟≫𝑠. In order not to affect spatial homogeneity on 𝑠 𝑥̂𝑎𝑥̂ scales comparable to or smaller than the electromagnetic 𝐸𝑎 =𝐵𝑎 ∼− 𝑖 (9) field, which propagates through the deconfining thermal 𝑖 𝑖 𝑟2 ground state in the absence of any explicit electric charges, 𝑙 and thus describes a static non-Abelian monopole of unit is considered a plane wave of wavelength much larger than 𝑠 electric and magnetic charges (dyon). For 𝑟≫𝑠≫𝛽(8) . Such a field effectively sees a density of self-dual dipoles; 𝑝𝑎 =𝑠𝛿𝑎 reduces to see (10). Because they are given by 𝑖 𝑖 their dipole moments align along the direction of the exciting electric or 𝛿𝑎 −3𝑥̂𝑎𝑥̂ 𝐸𝑎 =𝐵𝑎 ∼𝑠 𝑖 𝑖 . (10) magnetic field both in space and in the SU(2) algebra su(2). 𝑖 𝑖 𝑟3 Note that at this stage the definition of what is to be viewed as an Abelian direction in su(2) is a global gauge convention This is the field strength of a static, self-dual non-Abelian 𝑝𝑎 𝑝𝑎 such that all spatial directions of the dipole moment 𝑖 are dipole field, its dipole moment 𝑖 given as a priori thinkable. That is, dynamical Abelian projection of 𝑎 𝑎 thenon-Abeliansituationof(10)isowedtotheAbelianand 𝑝𝑖 =𝑠𝛿𝑖 . (11) dipole aligning nature of the exciting, massless field [18]. Up Interestingly, the same distance 𝑠, which sets the separation to global gauge transformations, this field exists because of between the charge centers of an Abelian magnetic monopole the adjoint Higgs mechanism invoked by the inert field 𝜙. 𝑉 |𝜙|−1 = and its antimonopole in a nontrivial-holonomy caloron, Per spatial coarse-graining volume cg of radius 𝑟 3 prescribes here for the case of trivial holonomy how small 𝜌=√Λ /2𝜋𝑇 with needstobeinordertoreducethenon-Abeliandipoleof(10) 4 󵄨 󵄨−3 to the non-Abelian monopole constituent; see (9). For the HS 𝑉 = 𝜋 󵄨𝜙󵄨 , 𝑎 𝑎 𝑎 𝑎 cg 󵄨 󵄨 (13) anticaloron one simply replaces 𝐸𝑖 =𝐵𝑖 by 𝐸𝑖 =−𝐵𝑖 in (8), 3 (9), and (10). Finally, let us remark that the condition 𝑠≫𝛽,whichis the center of a self-dual HS caloron and the center of required for (9) and (10) to be valid, is always satisfied for the an antiself-dual HS anticaloron [18] reside. Note the large −1 𝑠 caloron scale 𝜌∼|𝜙| which is relevant for the building of hierarchy between (the minimal spatial distance to the the thermal ground state in the deconfining phase of SU(2) center of a (anti)caloron, which allows us to identify the |𝜙|−1 Yang-Mills thermodynamics [18]. Namely, one has static, (anti)self-dual dipole) and the radius of the sphere 𝑉 defining cg, 2 𝑠 𝜌 2 𝜆3/2 𝜆3 =𝜋( ) =𝜋( ) = ≥ 212.3, (12) 𝑠 1 𝜆 3/2 𝛽 𝛽 2𝜋 4𝜋 = 𝜆3/2 ≥ 25.83 ( ) . 󵄨 󵄨−1 2 𝜆 (14) 󵄨𝜙󵄨 𝑐 where 𝜆 ≡ 2𝜋𝑇/Λ ≥𝜆𝑐 = 13.87. Iftheexcitingfieldiselectricthenitseestwice the electric 𝑝𝑎 4. Thermal Ground State as dipole 𝑖 (cancellation of magnetic dipole between caloron andanticaloron),andifitismagneticitseestwice the Induced by a Probe 𝑎 magnetic dipole 𝑝𝑖 (cancellation of electric dipole between E =−B ⇔−E = B The postulate that thermal photon propagation should be caloron and anticaloron, ). To be definite, described by an SU(2) rather than a U(1) gauge principle was letusdiscusstheelectriccaseindetail,characterisedbyan E put forward a decade ago and has undergone various levels of exciting Abelian field 𝑒.Themodulusoftheaccordingdipole D ‖ E investigation ever since; see [18, 31]. As a result, the associated density 𝑒 𝑒 is given as −4 Yang-Mills scale Λ ∼ 1.0638 × 10 eV is fixed by low- 󵄨 󵄨 2𝑠 3 𝜆 1/2 frequency observation of the Cosmic Microwave Background 󵄨D 󵄨 = = Λ2𝜆1/2 ( ) . 󵄨 𝑒󵄨 𝑉 4𝜋 𝑐 𝜆 (15) (CMB) [32] to correspond to the critical temperature for the cg 𝑐 deconfining-preconfining phase transition being the CMB’s present baseline temperature 𝑇0 = 2.725 K[18].This In classical electromagnetism the relation between the fields (2) E𝑒 and D𝑒 is prompted the name SU CMB.Inthefollowingwewouldlike to investigate in what sense the vacuum parameters of classi- D𝑒 =𝜖0E𝑒, cal electrodynamics, namely, the electric permittivity 𝜖0 and (16) Advances in Mathematical Physics 5

where To produce the measured value for 𝜖0 as in (17) the ratio 𝜉 𝑄 in (21) is required to be 𝜖 = 5.52703 × 107 0 (17) 𝑄󸀠 Vm 𝜉≡ = 19.56. (22) 𝑄 is the electric permittivity of the vacuum, and 𝑄 = 1.602 × −19 10 A s denotes the electron charge (unit of elementary Thus, compared to the electron charge, the charge unit charge), now both in SI units. associated with a (anti)self-dual non-Abelian dipole, residing 𝜌 in the thermal ground state, is gigantic. According to electromagnetism the energy density EM |E |=|B | Discussing 𝜇0, we could have proceeded in complete carried by an external electromagnetic wave with 𝑒 𝑒 −1 is analogy to the case of 𝜖0.(Itwouldbe𝜇0 defining the ratio between the modulus of the magnetic dipole density and the 1 1 1 1 𝜌 = (𝜖 E2 + B2) = (𝜖 + ) E2. magnetic flux density |B|.)Here,however,thecomparison EM 0 𝑒 𝑒 0 𝑒 (18) 2 𝜇0 2 𝜇0 between non-Abelian magnetic charge and an elementary, magnetic, and Abelian charge is not facilitated since the latter 2 In natural units we have 𝜖0𝜇0 =1/𝑐 =1, and therefore (to does not exist in electrodynamics. assume 𝜖0𝜇0 =1just represents a short cut, it would have Finally, let us see what the condition that the wavelength 𝑙 come out automatically if we had treated the magnetic case of the electromagnetic disturbance considered in this section 𝑠 (2) explicitly) 𝜇0 =1/𝜖0.Thus is much larger than implies when invoking SU CMB.One has 2 𝜌 =𝜖0E𝑒 . (19) 2 2 2 EM 𝜆𝑐 𝜆 𝑇 𝑙≫ ( ) = 1.1254 m ( ) . (23) E 𝜌 2Λ 𝜆𝑐 2.725 K The 𝑒-field dependence of EM is converted into a fictitious temperature dependence by demanding that the temperature Setting 𝑇=𝑇𝑐 = 2.725 K in (23), we obtain a lower bound on (2) of the thermal ground state of SU CMB adjusts itself so as to the wavelength of 𝑙min = 1.1254 m. 𝜌 accommodate EM,

3 4.2. Explicitly Induced Dipole Densities. Let us now discuss 𝜌 =4𝜋Λ𝑇⇐⇒ −1 EM the case 𝛽≪|𝜙| ≪𝑟≪𝑠.Torelyonthepresenceof 𝜙 𝑟 𝜆 𝜆 1/2 (20) the inert adjoint scalar field of the effective theory, needs 󵄨 󵄨 2 𝑐 |𝜙|−1 = 󵄨E𝑒󵄨 =Λ√2 ( ) . to be larger than the spatial coarse-graining scale 𝜖0 𝜆𝑐 3/2 3/2 (1/2𝜋)𝜆𝑐 (𝜆/𝜆𝑐) 𝛽 ≥ 8.22𝛽. Within the according regime −1 |𝜙| ≤𝑟≪𝑠of spatial distances from the caloron center Equation (20) generalises the thermal situation of ground- at (𝜏=0, x =0) an electromagnetic wave of wavelength state energy density of Section 3.2, where ground-state ther- 𝑙 sees the self-dual field of a static, non-Abelian monopole malisation is induced by a thermal ensemble of excitations, to of electric and magnetic charge as in (9) which is centered the case where the thermal ensemble is missing but the probe at x =0. A self-dual Abelian field strength 𝐸𝑖 =𝐵𝑖 of this field induces a fictitious temperature and energy density monopole is obtained [33] as to the ground state. Combining (15), (16), and (20) and 𝑎 𝑎 introducing the ratio 𝜉 between the non-Abelian monopole 𝜙 𝑎 𝜙 𝑎 󸀠 𝐸 =𝐵 = 𝐸 = 𝐵 𝑄 𝑖 𝑖 󵄨 󵄨 𝑖 󵄨 󵄨 𝑖 (24) charge in the dipole and the (Abelian) electron charge (in 󵄨𝜙󵄨 󵄨𝜙󵄨 natural units, the actual charge of the monopole constituents 𝑎 𝑎3 within the (anti)self-dual dipole is 1/𝑔,where𝑔 is the with the field 𝜙 gauged from unitary gauge 𝜙 = 2|𝜙|𝛿 𝑎 𝑎 undetermined fundamental gauge coupling. This is absorbed into “hedgehog” gauge 𝜙 =2|𝜙|𝑥̂ . The according gauge into 𝜉) 𝑄,weobtain transformation is give in terms of the group element Ω≡ (1/2)𝜓 −̂𝑘⋅𝜎 𝑖 (1/2)𝜓 𝜎 , (𝑖 = 1, 2, 3) −1 cos sin ,where 𝑖 ,arethe 𝜖0 [𝑄 (Vm) ] ̂ Pauli matrices, 𝑘≡(𝑒̂3 × 𝑥)/̂ sin 𝜃, 𝑒̂3 is the third vector of 𝜃≡∠(𝑒̂ , 𝑥)̂ 𝜓=𝜃 −1 1/2 an orthonormal basis of space, 3 ,and for 3 Λ[m ] 0≤𝜃≤𝜋−𝜖 𝜃=𝜋 = ( ) 𝜉√𝜖 [𝑄 ( )−1]⇐⇒ , which smoothly drops to zero at , √ Λ [ ] 0 Vm (21) and the limit 𝜖→0is understood [33]. For the monopole 32𝜋 eV 󸀠 field 𝐸𝑖 to be normalized to charge −2𝑄 one(thefactortwo −1 𝑄󸀠 9 Λ[m ] infrontofthemonopolecharge is due to a contribution 𝜖 [𝑄 ( )−1]= 𝜉2. 0 Vm 32𝜋2 Λ [ ] to the monopole field strength of the anticaloron identical to eV that of the caloron) thus has 󸀠 󸀠 Notice that 𝜖0 does not exhibit any temperature dependence 2𝑄 𝑥̂ 2𝑄 𝜇 𝑥̂ 𝐸 =𝐵 =− 𝑖 =− 0 𝑖 . and thus no dependence on the field strength E𝑒.Itisa 𝑖 𝑖 2 2 (25) 4𝜋𝜖0 𝑟 4𝜋 𝑟 universal constant. In particular, 𝜖0 does not relate to the state of fictitious ground-state thermalisation which would The electric or magnetic poles of (25) should independently associate with the rest frame of a local heat bath. react by harmonic and linear acceleration to the presence 6 Advances in Mathematical Physics

of an external electric or magnetic field E𝑒 or B𝑒,respec- 4.3. Discussion. In Sections 4.1 and 4.2 an analysis was tively, forming a monochromatic electromagnetic wave of performed to clarify to what extent the thermal ground state 𝜔=2𝜋/𝑙 x =0 (2) frequency .At one has of SU CMB can be regarded as the luminiferous aether, supporting the propagation of an external electromagnetic E𝑒 = E0 sin (𝜔𝑡) (26) wave (probe) of field strengths |E𝑒|=|B𝑒| and wavelength and readily derives (as in Thomson scattering) that the 𝑙 which, by itself, is not thermal. induceddipolemomentp,say,fortheelectriccase,isgiven Section 4.1 has focussed on wavelengths that are large −2 as compared to the distance 𝑠=𝜋|𝜙| /𝛽, very large compared −1 󸀠 2 to the resolution limit |𝜙| of the effective theory for decon- E𝑒 (2𝑄 ) (2) p =− . (27) fining SU CMB andevenmoresoonthescaleofinverse 𝑚𝜔2 𝛽 (2) temperature (see (12)) when (anti)calorons of SU CMB Interestingly, by virtue of (25) the squared charge of the pole, manifest themselves as static (anti)self-dual dipoles whose 󸀠 2 (2𝑄 ) ,cancelsoutinp because its mass 𝑚 carries an identical dipole moment is set by a fictitious temperature representing the intensity of the probe via (20). And indeed, in this case factor (only the electric (magnetic) monopole is linearly and 𝜖 𝜇 harmonically accelerated by the external electric (magnetic) vacuum permittivity 0 and permeability 0 turn out to be universal constants; see (21). When confronted with their field E𝑒 (B𝑒) and hence 𝑚 carries electric (magnetic) field energy only): experimental values the charges of the “constituent” non- Abelian monopoles in a dipole follow in units of electron 1 ∞ 1 2 ∞ 𝑑𝑟 𝑚= 𝜖 4𝜋 ∫ 𝑑𝑟𝑟2𝐸 𝐸 = (2𝑄󸀠) ∫ charge; see (22). 0 𝑖 𝑖 2 2 |𝜙|−1 8𝜋𝜖0 |𝜙|−1 𝑟 Equations (23) and (20) indicate that an uncertainty-like relation between field |E𝑒| strength and wavelength 𝑙 takes 1 󸀠 2 󵄨 󵄨 = (2𝑄 ) 󵄨𝜙󵄨 󳨐⇒ (28) place as follows: 8𝜋𝜖0 9 󵄨 󵄨4 8Λ 8𝜋𝜖 E 󵄨E 󵄨 𝑙−1 ≪ . p =− 0 𝑒 . 󵄨 𝑒󵄨 2 (34) 󵄨 󵄨 2 𝜖0 󵄨𝜙󵄨 𝜔 𝑉 p Therefore, the larger the probe intensity is, the longer its Again, the volume cg, which underlies the dipole moment by containing a caloron and an anticaloron center, is given by wavelength is required to be in order to be supported by (13), and we have thermal ground-state physics. Equation (34) is a condition 󵄨 󵄨 󵄨 󵄨2 with no reference to temperature and as such should be 󵄨 󵄨 |p| 󵄨E 󵄨 󵄨𝜙󵄨 󵄨D 󵄨 = =6𝜖 󵄨 𝑒󵄨 󵄨 󵄨 , regarded valid independently of the constraint that, thermo- 󵄨 𝑒󵄨 𝑉 0 𝜔2 (29) 𝜆≥𝜆 cg dynamically speaking, 𝑐. Things are different for wavelengths that are large on −1 −2 and therefore the scale |𝜙| but short on the scale 𝑠=𝜋|𝜙|/𝛽.This 󵄨 󵄨 󵄨 󵄨2 󵄨D𝑒󵄨 󵄨𝜙󵄨 case is investigated in Section 4.2. Then a (anti)caloron can 𝜖 ≡ 󵄨 󵄨 =6𝜖 . (30) 0 󵄨E 󵄨 0 𝜔2 no longer be viewed as a static, (anti)self-dual dipole but 󵄨 𝑒󵄨 rather is represented by a static, (anti)self-dual monopole. In (30) also the vacuum permittivity 𝜖0 cancels out, and we However, an attempt to consider dipole moments as induced are left with the condition dynamically by monopole shaking through the probe fields 󵄨 󵄨 𝜖 𝜇 𝜔=√6 󵄨𝜙󵄨 ⇐⇒ renders the definitions of vacuum parameters 0 and 0 󵄨 󵄨 meaningless; see (30). It does yield a fixation of the probe’s −1 1/2 (31) 𝑙 |𝜙| 2 󵄨 󵄨−1 2 𝜆 wavelength in terms of though; see (31). While the 𝑙=√ 𝜋 󵄨𝜙󵄨 = √ 𝜋Λ−1𝜆1/2 ( ) , 3 󵄨 󵄨 3 𝑐 𝜆 former situation is not surprising because single magnetic 𝑐 charges violate the Bianchi identities for the electromagnetic where temperature 𝑇 (or 𝜆), again, is set by the local field field strength tensor 𝐹𝜇] it is nontrivial that 𝑙 turns out to −1 strengths of the electromagnetic probe according to (18) and self-consistently satisfy the constraint that 𝑠≫𝑙>|𝜙| . (20). Let us see whether the second of (31) is consistent with Note that the minimal wavelengths 𝑙min = 1.1254 mand |𝜙|−1 ≤𝑟=𝑙≪𝑠 √ −1 1/2 . The former inequality is self-evident, and 𝑙min = 2/3𝜋Λ 𝜆𝑐 = 0.112 masobtainedinSections4.1 the latter follows from and 4.2, respectively, are off by a factor of ten only. 𝑠 3 𝜆3/2 𝜆 3/2 𝜆 3/2 = √ 𝑐 ( ) = 10.069 ( ) . (32) 𝑙 8 𝜋 𝜆𝑐 𝜆𝑐 5. Summary and Conclusions 𝜆=𝜆 By setting 𝑐 we obtain from (31) a minimal wavelength We have addressed the question how the concept of a thermal ground state of SU(2) , which in a fully thermalised 2 −1 1/2 CMB 𝑙 = √ 𝜋Λ 𝜆 = 0.112 . (33) situation coexists with a spectrum of partially massive min 3 𝑐 m (adjoint Higgs mechanism) thermal excitations of the same This wavelength is about a factor of ten smaller than the temperature, can be employed to understand the propagation lowest possible value as expressed by (23). of a nonthermal probe (monochromatic electromagnetic Advances in Mathematical Physics 7 wave) in vacuum, characterised by electric permittivity 𝜖0 and References magnetic permeability 𝜇0.Todothis,wehaveappealedtothe fact that the thermal ground state emerges by a spatial coarse [1]M.Born,W.Heisenberg,andP.Jordan,“ZurQuanten- graining over (anti)self-dual fundamental Yang-Mills fields mechanik. 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