Thermal Ground State and Nonthermal Probes

Thierry Grandou and Ralf Hofmann, Thermal Ground State and Nonthermal Probes, Advances in Mathematical Physics 2015, article ID 197197, 8 pages. Published at doi.org/10.1155/2015/197197; preprint at arXiv:1504.05923.

Reproduced under the Creative Commons Attribution licence printed in the article. Attribution: T. Grandou and R. Hofmann, Advances in Mathematical Physics 2015, article ID 197197, doi:10.1155/2015/197197. Display equations are restated in words, marked as such; nothing else is altered, and everything outside the article — the summary, the claims and this note — is the site’s own writing.

Abstract

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, effectively describing the pure quantum part of the thermal ground state in the induced quantum field theory. Here we show for the limit of zero holonomy how (anti)calorons associate a temperature independent electric permittivity and magnetic permeability to the thermal ground state of SU(2)CMB, the Yang-Mills theory conjectured to underlie the fundamental description of thermal photon gases.

1. Introduction

Quantum Mechanics is a highly efficient framework to describe the subatomic world, including coherence phenomena that extend to macroscopic length and time scales. The key quantity to describe deviations from classical behavior is Planck’s quantum of action, the reduced constant, which determines the fundamental interaction between charged matter and the electromagnetic field and thus also the shape of blackbody spectra by relating frequency and wave vector to particle-like energy and momentum and by appeal to Bose-Einstein statistics. In Quantum Mechanics, the reduced quantum of action sets the strength of multiplicative noncommutativity for a pair of canonically conjugate variables such as position and momentum, implying the respective uncertainty relations.

Despite being generally accepted as a universal constant of nature and in spite of the fact that we are able to efficiently compute quantum mechanical amplitudes and quantum statistical averages for a vast variety of processes in particle collisions, atoms, and molecules, extended condensed-matter systems, and astrophysical objects to match experiment and observation very well, one should remain curious concerning the principle mechanism that causes the emergence of a universal quantum of action. In [11, 12] it was argued that the irreconcilability of classical Euclidean and Minkowskian time evolution as expressed by a time-periodic SU(2) (anti)self-dual gauge field configuration — a (anti)caloron — whose action equals the reduced quantum of action and is associated with one unit of winding about a central spacetime point gives rise to indeterminism in the process it mediates. That each unit of action assigned to (anti)calorons of radius equal to the inverse modulus of the scalar field, which dominate the emergence of the thermal ground state, equals the reduced quantum of action follows from the value of the coupling in the induced, effective, thermal quantum field theory [13–17] of the deconfining phase in SU(2) Yang-Mills thermodynamics. The coupling, in turn, obeys an evolution in temperature (flat almost everywhere) which represents the validity of Legendre transformations in the effective ensemble where the thermal ground state coexists with massive (adjoint Higgs mechanism) and massless (intact U(1)) thermal fluctuations. The thermal ground state thus is a spatially homogeneous ensemble of quantum fluctuations carried by (anti)caloron centers. At the same time, as we will see, this state provides electric and magnetic dipole densities supporting the propagation of certain electromagnetic waves in an SU(2) Yang-Mills theory of scale about ten to the minus fourth electron volts, SU(2)CMB [18].

In the present work, we establish this link between quantised action, represented by the scalar field, and classical wave propagation enabled by the vacuum parameters — the electric permittivity and magnetic permeability of the vacuum — in terms of the central and peripheral structure of a trivial-holonomy (anti)caloron, respectively. That is, by allowing a fictitious temperature to represent the energy density of an electromagnetic wave (nonthermal, external probe) via the thermal ground state through which it propagates we ask what this implies for the permittivity and the permeability. As a result, both neither depend on the temperature nor, as we will argue, on a singled-out inertial frame provided that a certain condition on intensity and frequency of the wave is satisfied. This is a first step in reviving the concept of the luminiferous aether, albeit now with the goal of constructing a Poincaré invariant object.

This paper is organised as follows. In the next section we shortly discuss key features of the effective theory for the deconfining phase of SU(2) Yang-Mills thermodynamics. Section 3 contains a reminder to principles in interpreting a Euclidean field configuration in terms of Minkowskian observables. In a next step, general facts are reviewed on Euclidean, periodic, (anti)self-dual field configurations of charge modulus unity concerning the central locus of action, their holonomy, and their behaviour under semiclassical deformation. Finally, we review the anatomy of a zero-holonomy Harrington-Shepard (HS) caloron in detail, pointing out its staticity for spatial distances from the center that exceed the inverse of temperature, and discuss which static charge configuration it resembles depending on two distinct spatial distance regimes. In Section 4 we briefly review the postulate that an SU(2) Yang-Mills theory of scale about ten to the minus fourth electron volts, SU(2)CMB, describes thermal photon gases [18]. Subsequently, the large-distance regime in an HS (anti)caloron is considered in order to deduce an expression for the permittivity based on knowledge about the electric dipole moment provided by a (anti)caloron of radius equal to the inverse modulus of the scalar field, the size of the spatial coarse-graining volume, and the fact that the energy density of the probe must match that of the thermal ground state. As a result, the permittivity and the permeability turn out to be independent of temperature, the former representing an electric charge, large on the scale of the electron charge, of the fictitious constituent monopoles giving rise to the associated dipole density. Zooming in to smaller spatial distances to the center, the HS (anti)caloron exhibits isolated (anti)self-dual monopoles. For them to turn into dipoles shaking by the probe fields is required. We then show that the definitions of the permittivity and the permeability, which were successfully applied to the large-distance regime, become meaningless. Finally, our results are discussed. Section 5 summarises the paper and discusses how a universality of the permittivity and the permeability for the entire, experimentally investigated electromagnetic spectrum could emerge.

2. Sketch of Deconfining SU(2) Yang-Mills Thermodynamics

For deconfining SU(2) Yang-Mills thermodynamics, a spatial coarse graining over the (anti)self-dual, that is, the nonpropagating [19], topological sector with charge modulus one can be performed; see [18] and references therein, to yield an inert adjoint scalar field. Its modulus sets the maximal possible resolution in the effective theory whose ground state energy density essentially is given as four pi times the cube of the Yang-Mills scale times the temperature — the scale being a constant of integration of dimension mass — and whose propagating sector is, in a totally fixed, physical gauge (unitary-Coulomb) characterised by a massless mode (the photon, unbroken U(1) subgroup of SU(2)) and two thermal quasiparticle modes of equal mass, twice the coupling times the modulus of the scalar field, with mass induced by the adjoint Higgs mechanism, which propagate thermally, that is, on-shell only. Interactions within this propagating sector are mediated by isolated (anti)calorons whose action is argued to be the reduced quantum of action [11, 12]. Judged in terms of inclusive quantities such as radiative corrections to the one-loop pressure or the energy density of blackbody radiation, these interactions are feeble, and their expansion into one-particle-irreducible bubble diagrams is conjectured to terminate at a finite number of loops [18]. However, spectrally seen, the effects of the massive modes interacting with the photon lead to severe consequences at low frequencies and temperatures comparable to the critical temperature where screened (anti)monopoles, released by (anti)caloron dissociation upon large-holonomy deformations [20], rapidly become massless and thus start to condense.

3. Caloron Structure

3.1. Euclidean Field Theory and Interpretable Quantities

Nontrivial solutions to an elliptic differential equation, such as the Euclidean Yang-Mills equation, no longer are solutions of the corresponding hyperbolic equation upon analytic continuation of Euclidean time into imaginary Minkowskian time (Wick rotation). To endow meaning to quantities computed on classical field configurations on a four-dimensional Euclidean spacetime in SU(2) Yang-Mills thermodynamics in terms of observables in a Minkowskian spacetime we thus must insist that these quantities are not affected by the Wick rotation. That is, to assign a real-world interpretation to a Euclidean quantity it needs to be (i) either stationary (not depend on Euclidean time) or (ii) associated with an instant in Euclidean spacetime because, by exploiting time translational invariance of the Yang-Mills action, this instant can be picked as the origin of Euclidean time at a given point in space.

3.2. Review of General Facts

If not stated otherwise we work in supernatural units, in which the reduced quantum of action, the speed of light in vacuum and Boltzmann’s constant are all set to one. A trivial-holonomy caloron of topological charge unity on the cylinder formed by a circle and three-dimensional space, where the circle has circumference equal to the inverse temperature and describes the compactified Euclidean time dimension, is constructed by an appropriate superposition of charge-one singular-gauge instanton prepotentials [21, 22] with the temporal coordinate of their instanton centers equidistantly stacked along the infinitely extended Euclidean time dimension [23] to enforce temporal periodicity of the gauge field.

(Equations 1 and 2, restated in words: for gauge group SU(2) the Harrington-Shepard caloron gauge field is built from the antiself-dual ’t Hooft symbol contracted with the antihermitian generators and the derivative of the logarithm of a profile function. That profile function equals one, plus pi times the squared scale parameter divided by the product of the inverse temperature and the radial distance, multiplied by the hyperbolic sine of two pi times the radial distance over the inverse temperature, divided by the difference between the hyperbolic cosine of that same argument and the cosine of two pi times Euclidean time over the inverse temperature.)

Here the scale parameter is that of the singular-gauge instanton used to seed the “mirror sum”, leading to equation 2. The associated antiself-dual field configuration is obtained by replacing the antiself-dual ’t Hooft symbol by the self-dual one in equation 1.

Configuration (1) is singular where both Euclidean time and radial distance vanish. This point is the locus of the configuration’s topological charge of unity in the sense that the integral of the Chern-Simons current over a three-sphere of any radius centered there yields unity independently of that radius. Self-duality implies that the action of the HS caloron is given as

(Equation 3, restated in words: the caloron action equals eight pi squared divided by the square of the Euclidean coupling constant, obtained as the same factor multiplying the integral of the Chern-Simons current over the three-sphere.)

where the coupling is that of the Euclidean (classical) theory. Equation (3) holds in the limit of vanishing sphere radius, meaning that the action can be attributed to the singularity of the HS solution at the origin and thus has a Minkowskian interpretation; see Section 3.1. Based on [13–16] and on the fact that the thermal ground state emerges from unit-charge calorons and anticalorons, whose scale parameter essentially coincides with the inverse of maximal resolution in the effective theory for deconfining SU(2) Yang-Mills thermodynamics, it was argued in [11] (see also [12]) that the caloron action, as well as the action of an HS anticaloron of the same scale, equals the reduced quantum of action if the effective theory is to be interpreted as a local quantum field theory.

The HS caloron is the trivial-holonomy limit of the self-dual Lee-Lu-Kraan-van-Baal (LLKvB) configuration with unit charge and total magnetic charge zero [24–26] which is constructed via the Nahm transformation of self-dual fields on the Euclidean four torus [27–29]. For nontrivial holonomy the LLKvB solution exhibits a pair of a magnetic monopole and its antimonopole with respect to the Abelian subgroup U(1) inside SU(2) left unbroken by the asymptotic value of the temporal gauge field component. Their masses are four pi times the holonomy parameter and four pi times the difference between two pi over the inverse temperature and the holonomy parameter, such that in the trivial-holonomy limits one of these magnetic constituents becomes massless and thus completely spatially delocalised. For nontrivial holonomy, where both monopole and antimonopole are of finite mass, localised, and separated by a spatial distance

(Equation 4, restated in words: the separation equals pi times the squared caloron scale parameter divided by the inverse temperature.)

they can be considered static by an exact cancellation of attraction, mediated by their U(1) magnetic fields, and repulsion due to the temporal gauge field component. As was shown in [20] by investigating the effective action of a LLKvB caloron (integrating out Gaussian fluctuations), this balance is distorted, leading to monopole-antimonopole attraction for small holonomy in two windows at either end of the holonomy range

(Equation 5, restated in words: attraction holds for holonomy up to one over the square root of three, subtracted from one and multiplied by pi over the inverse temperature, and again for holonomy above the corresponding value with a plus sign, up to two pi over the inverse temperature.)

and to repulsion in the complementary range of (large) holonomy. Because there is no localised counterpart to a monopole or antimonopole in the trivial-holonomy limit, HS calorons must be considered stable under Gaussian fluctuations, in contrast to the case of nontrivial holonomy which is unstable. The latter statement is also mirrored by the fact that a nontrivial, static holonomy leads to zero quantum weight in the infinite-volume limit, which is realistic at high temperatures [18] where the radius of the spatial coarse-graining volume for a single caloron diverges as the square root of two pi times the temperature over the cube of the Yang-Mills scale. As a consequence, nontrivial holonomy can only occur transiently in configurations which do not saturate (anti)self-duality bounds to the Yang-Mills action. Again, this is equivalent to stating the instability of the LLKvB solution. It can be shown [18] that the small-holonomy case of monopole-antimonopole attraction by far dominates the situation of monopole-antimonopole repulsion when a caloron dissociates into its constituents.

The spatial coarse graining over (anti)self-dual calorons of charge modulus one, which do not propagate (due to (anti)self-duality their energy-momentum tensor vanishes identically [19]), yielding a highly accurate a priori estimate of the deconfining thermal ground state in terms of an inert, adjoint scalar field and a pure-gauge configuration, is performed over isolated and stable HS solutions [18]. The coarse-grained pure-gauge field represents a posteriori the effects of small holonomy changes due to (anti)caloron overlap and interaction.

3.3. Anatomy of a Relevant Harrington-Shepard Caloron

Let us now review [30] how the field strength of an HS caloron depends on the distance from its center. For four-dimensional distances much smaller than the inverse temperature one has

(Equation 6, restated in words: the profile function is approximated by one plus pi times the separation over three times the inverse temperature, plus the squared scale parameter over the squared four-distance, up to corrections of order the squared four-distance over the squared inverse temperature.)

where the separation is that defined in equation 4. From (6) and (1) one obtains for four-dimensional distances much smaller than the inverse temperature an expression for the self-dual field strength.

(Equation 7 is written in index notation with the ’t Hooft symbol and an inversion tensor; in words, the field strength at small four-dimensional distances from the caloron center behaves like that of a singular-gauge instanton with a renormalised scale parameter — the original squared scale parameter divided by one plus pi over three times the separation over the inverse temperature.)

Therefore, the field strength of the HS solution exhibits a dependence on Euclidean time and as such has no Minkowskian interpretation; see Section 3.1.

For a Minkowskian spacetime one can infer, however, that the action of the configuration is attributable to winding of the caloron around the group manifold as induced by a spacetime point, the instanton center. This is because, in the sense of (3), an instant has no analytic continuation or Wick rotation. (The four-dimensional action or topological-charge density of the caloron is regular at the center, does depend on Euclidean spacetime in the vicinity of this point, and thus has no Minkowskian interpretation.)

For radial distances much larger than the inverse temperature the self-dual electric and magnetic fields are static and can be written as

(Equation 8, restated in words: electric and magnetic field components are equal, and are given by minus the quantity formed from the product of the two unit direction vectors divided by the squared radius, minus one over the product of radius and separation multiplied by the difference between the Kronecker delta and three times the product of unit direction vectors, all divided by the square of one plus the radius over the separation.)

For radial distances between the inverse temperature and the separation, equation 8 simplifies as

(Equation 9, restated in words: the electric and magnetic fields both equal minus the product of the two unit direction vectors divided by the squared radius.)

and thus describes a static non-Abelian monopole of unit electric and magnetic charges (dyon). For radial distances much larger than the separation, which in turn is much larger than the inverse temperature, equation 8 reduces to

(Equation 10, restated in words: the electric and magnetic fields both equal the separation multiplied by the difference between the Kronecker delta and three times the product of the unit direction vectors, divided by the cube of the radius.)

This is the field strength of a static, self-dual non-Abelian dipole field, its dipole moment given as

(Equation 11, restated in words: the dipole moment equals the separation multiplied by the Kronecker delta linking the spatial and algebra directions.)

Interestingly, the same distance which sets the separation between the charge centers of an Abelian magnetic monopole and its antimonopole in a nontrivial-holonomy caloron prescribes here for the case of trivial holonomy how small the radius needs to be in order to reduce the non-Abelian dipole of (10) to the non-Abelian monopole constituent; see (9). For the HS anticaloron one simply reverses the sign of the magnetic field relative to the electric field in equations 8, 9 and 10.

Finally, let us remark that the condition that the separation be much larger than the inverse temperature, which is required for (9) and (10) to be valid, is always satisfied for the caloron scale relevant for the building of the thermal ground state in the deconfining phase of SU(2) Yang-Mills thermodynamics [18]. Namely, one has

(Equation 12, restated in words: the ratio of separation to inverse temperature equals pi times the squared ratio of scale parameter to inverse temperature, which equals the cube of the dimensionless temperature divided by four pi, and is at least 212.3)

where the dimensionless temperature is two pi times the temperature over the Yang-Mills scale and is bounded below by its critical value of 13.87.

4. Thermal Ground State as Induced by a Probe

The postulate that thermal photon propagation should be described by an SU(2) rather than a U(1) gauge principle was put forward a decade ago and has undergone various levels of investigation ever since; see [18, 31]. As a result, the associated Yang-Mills scale of about 1.0638 times ten to the minus fourth electron volts is fixed by low-frequency observation of the Cosmic Microwave Background (CMB) [32] to correspond to the critical temperature for the deconfining-preconfining phase transition being the CMB’s present baseline temperature of 2.725 K [18]. This prompted the name SU(2)CMB. In the following we would like to investigate in what sense the vacuum parameters of classical electrodynamics, namely, the electric permittivity and the magnetic permeability, can be reduced to the physics of the static, non-Abelian, and (anti)self-dual monopole and dipole configurations represented by HS (anti)calorons in the two distance regimes; see Section 3.3. To do this, the concept of a thermal ground state together with information on how it is obtained [18] as well as the results of Section 3.3 [30] are invoked.

4.1. Preexisting Dipole Densities

Let us discuss the case where the radius is much larger than the separation. In order not to affect spatial homogeneity on scales comparable to or smaller than the separation the electromagnetic field, which propagates through the deconfining thermal ground state in the absence of any explicit electric charges, is considered a plane wave of wavelength much larger than that separation. Such a field effectively sees a density of self-dual dipoles; see (10). Because their dipole moments are proportional to the Kronecker delta, they align along the direction of the exciting electric or magnetic field both in space and in the SU(2) algebra. Note that at this stage the definition of what is to be viewed as an Abelian direction in the algebra is a global gauge convention such that all spatial directions of the dipole moment are a priori thinkable. That is, dynamical Abelian projection of the non-Abelian situation of (10) is owed to the Abelian and dipole aligning nature of the exciting, massless field [18]. Up to global gauge transformations, this field exists because of the adjoint Higgs mechanism invoked by the inert scalar field.

Per spatial coarse-graining volume of radius equal to the inverse modulus of the scalar field, which equals the square root of the cube of the Yang-Mills scale over two pi times the temperature, with

(Equation 13, restated in words: the coarse-graining volume is four thirds pi times the cube of the inverse modulus of the scalar field.)

the center of a self-dual HS caloron and the center of an antiself-dual HS anticaloron [18] reside. Note the large hierarchy between the separation — the minimal spatial distance to the center of a (anti)caloron which allows us to identify the static, (anti)self-dual dipole — and the radius of the sphere defining the coarse-graining volume,

(Equation 14, restated in words: the ratio of the separation to the inverse modulus of the scalar field is one half of the three-halves power of the dimensionless temperature, which is at least 25.83 times the three-halves power of the temperature in units of its critical value.)

If the exciting field is electric then it sees twice the electric dipole (cancellation of magnetic dipole between caloron and anticaloron), and if it is magnetic it sees twice the magnetic dipole (cancellation of electric dipole between caloron and anticaloron). To be definite, let us discuss the electric case in detail, characterised by an exciting Abelian electric field. The modulus of the according dipole density, parallel to that field, is given as

(Equation 15, restated in words: the dipole density equals twice the separation divided by the coarse-graining volume, which equals three over four pi times the squared Yang-Mills scale times the square root of the critical dimensionless temperature, multiplied by the square root of the ratio of critical to actual dimensionless temperature.)

In classical electromagnetism the relation between the electric field and the dipole density is

(Equation 16, restated in words: the dipole density equals the vacuum permittivity multiplied by the electric field.)

where

(Equation 17, restated in words: the electric permittivity of the vacuum equals 5.52703 times ten to the seventh elementary charges per volt metre.)

is the electric permittivity of the vacuum, and the elementary charge is 1.602 times ten to the minus nineteenth ampere seconds, now both in SI units.

According to electromagnetism the energy density carried by an external electromagnetic wave with equal electric and magnetic field moduli is

(Equation 18, restated in words: the energy density is one half of the sum of the permittivity times the squared electric field and the squared magnetic field over the permeability, which equals one half of the sum of the permittivity and the inverse permeability, multiplied by the squared electric field.)

In natural units the product of permittivity and permeability equals the inverse squared speed of light, which is one, and therefore the permeability is the reciprocal of the permittivity. (To assume that product equals one just represents a short cut; it would have come out automatically if we had treated the magnetic case explicitly.) Thus

(Equation 19, restated in words: the energy density of the probe equals the permittivity multiplied by the squared electric field.)

The field dependence of the probe energy density is converted into a fictitious temperature dependence by demanding that the temperature of the thermal ground state of SU(2)CMB adjusts itself so as to accommodate that energy density,

(Equation 20, restated in words: setting the probe energy density equal to four pi times the cube of the Yang-Mills scale times the temperature gives the electric field modulus as the squared Yang-Mills scale times the square root of two over the permittivity, times the square root of the critical dimensionless temperature, times the square root of the ratio of actual to critical dimensionless temperature.)

Equation (20) generalises the thermal situation of ground-state energy density of Section 3.2, where ground-state thermalisation is induced by a thermal ensemble of excitations, to the case where the thermal ensemble is missing but the probe field induces a fictitious temperature and energy density to the ground state. Combining (15), (16), and (20) and introducing the ratio between the non-Abelian monopole charge in the dipole and the Abelian electron charge — in natural units the actual charge of the monopole constituents within the (anti)self-dual dipole is the reciprocal of the undetermined fundamental gauge coupling, and this is absorbed into that ratio — we obtain

(Equation 21, restated in words: the permittivity in elementary charges per volt metre equals nine over thirty-two pi squared, multiplied by the ratio of the Yang-Mills scale expressed in inverse metres to the same scale expressed in electron volts, multiplied by the square of the charge ratio.)

Notice that the permittivity does not exhibit any temperature dependence and thus no dependence on the field strength. It is a universal constant. In particular, it does not relate to the state of fictitious ground-state thermalisation which would associate with the rest frame of a local heat bath.

To produce the measured value for the permittivity as in (17) the charge ratio in (21) is required to be

(Equation 22, restated in words: the ratio of the non-Abelian monopole charge to the electron charge equals 19.56.)

Thus, compared to the electron charge, the charge unit associated with a (anti)self-dual non-Abelian dipole, residing in the thermal ground state, is gigantic.

Discussing the permeability, we could have proceeded in complete analogy to the case of the permittivity. (It would be the inverse permeability defining the ratio between the modulus of the magnetic dipole density and the magnetic flux density.) Here, however, the comparison between non-Abelian magnetic charge and an elementary, magnetic, and Abelian charge is not facilitated since the latter does not exist in electrodynamics.

Finally, let us see what the condition that the wavelength of the electromagnetic disturbance considered in this section is much larger than the separation implies when invoking SU(2)CMB. One has

(Equation 23, restated in words: the wavelength must be much larger than the squared critical dimensionless temperature over twice the Yang-Mills scale, multiplied by the squared ratio of actual to critical dimensionless temperature, which equals 1.1254 metres multiplied by the square of the temperature in units of 2.725 kelvin.)

Setting the temperature equal to the critical value in (23), we obtain a lower bound on the wavelength of 1.1254 m.

4.2. Explicitly Induced Dipole Densities

Let us now discuss the case where the radius lies between the coarse-graining scale and the separation. To rely on the presence of the inert adjoint scalar field of the effective theory, the radius needs to be larger than the spatial coarse-graining scale, which is at least 8.22 times the inverse temperature. Within the according regime of spatial distances from the caloron center an electromagnetic wave of a given wavelength sees the self-dual field of a static, non-Abelian monopole of electric and magnetic charge as in (9) which is centered at the origin. A self-dual Abelian field strength of this monopole is obtained [33] as

(Equation 24, restated in words: the Abelian electric field equals the Abelian magnetic field, and both are obtained by projecting the non-Abelian field strength onto the direction of the scalar field in the algebra.)

with the scalar field gauged from unitary gauge into “hedgehog” gauge. The according gauge transformation is given in terms of a group element built from the cosine and sine of half the polar angle together with the Pauli matrices and a unit vector formed from the cross product of the third basis vector with the radial unit vector, the angle smoothly dropping to zero at the pole [33]. For the monopole field to be normalized to twice the negative monopole charge — the factor two in front of the monopole charge is due to a contribution to the monopole field strength of the anticaloron identical to that of the caloron — one thus has

(Equation 25, restated in words: the Abelian electric and magnetic fields both equal minus twice the monopole charge divided by four pi times the permittivity times the squared radius, times the radial unit vector; equivalently minus twice the monopole charge times the permeability over four pi times the squared radius, times the radial unit vector.)

The electric or magnetic poles of (25) should independently react by harmonic and linear acceleration to the presence of an external electric or magnetic field, respectively, forming a monochromatic electromagnetic wave of angular frequency two pi over the wavelength. At the origin one has

(Equation 26, restated in words: the exciting electric field is an amplitude multiplied by the sine of the angular frequency times time.)

and one readily derives (as in Thomson scattering) that the induced dipole moment, say, for the electric case, is given as

(Equation 27, restated in words: the induced dipole moment equals minus the exciting field multiplied by the square of twice the monopole charge, divided by the mass times the squared angular frequency.)

Interestingly, by virtue of (25) the squared charge of the pole cancels out in the dipole moment because its mass carries an identical factor. Only the electric (magnetic) monopole is linearly and harmonically accelerated by the external electric (magnetic) field, and hence the mass carries electric (magnetic) field energy only:

(Equation 28, restated in words: the mass is one half of the permittivity times four pi times the radial integral of the squared monopole field from the coarse-graining scale outward, which equals the square of twice the monopole charge divided by eight pi times the permittivity times the coarse-graining scale; substituting this into equation 27 gives the induced dipole moment as minus eight pi times the permittivity times the exciting field divided by the modulus of the scalar field and the squared angular frequency.)

Again, the coarse-graining volume, which underlies the dipole moment by containing a caloron and an anticaloron center, is given by (13), and we have

(Equation 29, restated in words: the induced dipole density equals the dipole moment over the coarse-graining volume, which equals six times the permittivity multiplied by the exciting field and the squared modulus of the scalar field, divided by the squared angular frequency.)

and therefore

(Equation 30, restated in words: the permittivity, defined as the ratio of dipole density to exciting field, equals six times the permittivity multiplied by the squared modulus of the scalar field over the squared angular frequency — an identity in which the permittivity cancels.)

In (30) also the vacuum permittivity cancels out, and we are left with the condition

(Equation 31, restated in words: the angular frequency equals the square root of six times the modulus of the scalar field; equivalently the wavelength equals the square root of two thirds times pi times the inverse Yang-Mills scale times the square root of the critical dimensionless temperature, multiplied by the square root of the ratio of actual to critical dimensionless temperature.)

where the temperature, again, is set by the local field strengths of the electromagnetic probe according to (18) and (20). Let us see whether the second of (31) is consistent with the requirement that the wavelength lie between the coarse-graining scale and the separation. The former inequality is self-evident, and the latter follows from

(Equation 32, restated in words: the ratio of separation to wavelength equals the square root of three eighths times the three-halves power of the critical dimensionless temperature over pi, times the three-halves power of the temperature in units of its critical value, which equals 10.069 times that same power.)

By setting the temperature equal to its critical value we obtain from (31) a minimal wavelength

(Equation 33, restated in words: the minimal wavelength equals the square root of two thirds times pi times the inverse Yang-Mills scale times the square root of the critical dimensionless temperature, which equals 0.112 metres.)

This wavelength is about a factor of ten smaller than the lowest possible value as expressed by (23).

4.3. Discussion

In Sections 4.1 and 4.2 an analysis was performed to clarify to what extent the thermal ground state of SU(2)CMB can be regarded as the luminiferous aether, supporting the propagation of an external electromagnetic wave (probe) of equal electric and magnetic field strengths and a given wavelength which, by itself, is not thermal.

Section 4.1 has focussed on wavelengths that are large compared to the separation, very large compared to the resolution limit of the effective theory for deconfining SU(2)CMB and even more so on the scale of inverse temperature (see (12)) when (anti)calorons of SU(2)CMB manifest themselves as static (anti)self-dual dipoles whose dipole moment is set by a fictitious temperature representing the intensity of the probe via (20). And indeed, in this case vacuum permittivity and permeability turn out to be universal constants; see (21). When confronted with their experimental values the charges of the “constituent” non-Abelian monopoles in a dipole follow in units of electron charge; see (22).

Equations (23) and (20) indicate that an uncertainty-like relation between field strength and wavelength takes place as follows:

(Equation 34, restated in words: the fourth power of the exciting field modulus divided by the wavelength must be much smaller than eight times the ninth power of the Yang-Mills scale divided by the squared permittivity.)

Therefore, the larger the probe intensity is, the longer its wavelength is required to be in order to be supported by thermal ground-state physics. Equation (34) is a condition with no reference to temperature and as such should be regarded valid independently of the constraint that, thermodynamically speaking, the dimensionless temperature is at or above its critical value.

Things are different for wavelengths that are large on the scale of the resolution limit but short on the scale of the separation. This case is investigated in Section 4.2. Then a (anti)caloron can no longer be viewed as a static, (anti)self-dual dipole but rather is represented by a static, (anti)self-dual monopole. However, an attempt to consider dipole moments as induced dynamically by monopole shaking through the probe fields renders the definitions of vacuum parameters meaningless; see (30). It does yield a fixation of the probe’s wavelength in terms of the resolution limit though; see (31). While the former situation is not surprising because single magnetic charges violate the Bianchi identities for the electromagnetic field strength tensor it is nontrivial that the wavelength turns out to self-consistently satisfy the constraint that it lie between the resolution limit and the separation. Note that the minimal wavelengths of 1.1254 m and 0.112 m as obtained in Sections 4.1 and 4.2, respectively, are off by a factor of ten only.

5. Summary and Conclusions

We have addressed the question how the concept of a thermal ground state of SU(2)CMB, which in a fully thermalised situation coexists with a spectrum of partially massive (adjoint Higgs mechanism) thermal excitations of the same temperature, can be employed to understand the propagation of a nonthermal probe (monochromatic electromagnetic wave) in vacuum, characterised by electric permittivity and magnetic permeability. To do this, we have appealed to the fact that the thermal ground state emerges by a spatial coarse graining over (anti)self-dual fundamental Yang-Mills fields of topological charge modulus unity at finite temperature: Harrington-Shepard (anti)calorons of trivial holonomy. Note that this coarse graining does not require the consideration of thermal excitations. Therefore, it is suggestive that the concept of the thermal ground state can be extended to the description of a nonthermal situation with the temperature acting as a period in compactified Euclidean spacetime and no longer as a thermodynamical temperature.

Knowing how large the coarse-graining volume is, which contains one caloron and one anticaloron center, where the fundamental unit of action is localised (Section 3.2), and by exploiting the structure of these field configurations spatially far away (Section 3.3) from their centers, we were able to deduce densities of electric and magnetic dipoles in Section 4.1. Dividing these dipole densities by the respective field strengths of the probe, self-consistently adjusted to the energy density of the thermal ground state (small, transient (anti)caloron holonomies), yields definitions of the permittivity and the permeability. In the electric case a match with the experimental value predicts the charge of one of the monopoles, which constitutes the dipole, in terms of electron charge. The former charge turns out to be substantially larger than the latter.

As shown in Section 4.2 this way of reasoning, which is valid for large wavelengths only, cannot be extended to smaller wavelengths. Namely, in a region of spatial distances to the (anti)caloron center, where the configuration resembles (anti)self-dual, static monopoles, the definition of the permittivity and the permeability in terms of dipole densities that are explicitly induced by the probe’s oscillating field strengths becomes meaningless. This is expected since the existence of resolved magnetic monopoles would violate the Bianchi identities for the field strength tensor of electromagnetism.

We conclude that the thermal ground state of SU(2)CMB supports the propagation of a nonthermal probe purely in terms of Harrington-Shepard (anti)calorons (trivial holonomy) if an uncertainty-like relation between wavelength and the square of the probe’s intensity is obeyed; see (34).

To address the nonthermal propagation of shorter wavelength and/or higher intensities (see (34)), additional, mixing SU(2) gauge factors of hierarchically larger Yang-Mills scales have to be postulated; see discussion in [34]. At present, however, it is not clear how the effectiveness of the very successful Standard Model of particle physics in describing electroweak processes can be achieved in terms of such a more fundamental framework of pure Yang-Mills dynamics.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

One of us (Ralf Hofmann) would like to thank Stan Brodsky for a stimulating conversation.

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(End of the reproduced article. On this site, Hofmann’s nonperturbative approach to Yang-Mills thermodynamics is at /library/stm-fb901bd417 and his SU(2) and SU(3) implications paper at /library/stm-468ce7295f; the centre-vortex loops that carry the confining phase are at /library/stm-e1f7b72444; the hot-spot evaporation spectra are at /library/stm-dcaf382795; the axial anomaly applied to galaxies and the dark universe is at /library/stm-b06591e288; and the modified MIT bag vacuum structure is at /library/stm-6fa6fa9f58.)