Gauge Symmetry
The exponential function for the coupling energy e3vΦ, as derived from experiment (1), represents a capturing of the magnetic condensate wavefunction through the energy degeneracy of the water ice cage structures such that it becomes real and observable. It also expresses the scale-invariant and gauge-invariant properties of the critical system. Conservation of angular momentum requires the existence of a sink/ source for the associated changes in inertia (variable effective radius) together with a corresponding symmetry relation.
Scale-invariance is attributed to the hyperbolic curvature of Lorentz boosts that impose a conformal symmetry on the embedding vacuum manifold [
24]. Conformal symmetry is able to describe the tetrahedral, hydrogen bonded, 3-dimensional spatial geometry of the crystal-fluid under non-extensive volume changes whilst its variable hyperbolic surface area maps to the gradient energy term of the Lagrangian (see below). Since a universality class of topologically invariant critical exponents has been determined for the continuous (second order) phase transition, the system can be modelled through conformal field theory in 4-spacetime dimensions, ie. it is describable by a renormalizable quantum field theory in which the non-perturbative conformal bootstrap is irrelevant [
25].
Yang-Mills theory is a strongly coupled quantum field theory [
18], ie. a gauge theory in which the low-energy dynamics are far removed from any classical description [
26]. It is represented through the mathematical structure of Lie groups that provide for intricate topologies. The compact, simple Lie group
SU(3) describes the strong interaction in QCD, ie. the binding of quarks and gluons through confinement mechanisms. The mechanical action of the piston expander can be described by the emergence of a gauge field
Φ and the critical length exponent
v (as Equation (4) below). In QCD such gauge fields are collectively known as gluon fields. The field strength, or curvature
Fμν, has the general form:
where
Aν provides for Lorentz invariance and
Aμ is the gauge connection.
The gauge connection depends upon a complex scaling symmetry that is exact but not directly observable [
27]. In the quantum state
Ψ →
eiθΨ, which could be interpreted as a potential sink/ source for the ‘hidden’ inertia of the false vacuum system (although later this is revealed not to be the case). It also represents the complex order parameter field of the Ginzburg-Landau superconducting phase transition included in Equation (7) below.
Experimental results lead to a relativistic manifestation of length expansion and time contraction arising from false vacuum behaviour in a thermodynamically constrained condensed matter system. The local stability conditions maintained through dynamically responsive inhomogeneities in this soft matter are deemed equivalent to the property of asymptotic freedom, or antiscreening, which accounts for the mechanism of colour confinement in particle physics, ie. scale-invariance is effective across the micro- and macro-scales of the system. In QCD it is the emergence of clouds of virtual gluons that establish the antiscreening phenomenon [
5]. In both mechanisms, increasing kinetic energy is mirrored by an increasing negative energy potential such that total energy remains constant.
Whilst the crystal-fluid material displays high stability in total energy and density, the embedding manifold always remains on the threshold of instability. Small positive or negative pressure perturbations produce divergent critical behaviour manifesting as large variations in swept volume V. However, this is not the specific volume of the material system (density remaining almost constant) but rather the non-extensive volume change associated with the condensation of magnetic charges and simultaneous emergence of a gradient energy term.
The ‘rolling’ critical response initiated by anisotropy in water ice cage structures facilitates net energy gain for the duration of external pressure perturbations, either positive or negative, in a display of self-organized criticality [
28]. The angular momentum of the material is transferred to or from the embedding vacuum manifold through self-organizing behaviour and high energy degeneracy of the water ice cage structures. However, this brief statement does not provide a full description and a more detailed hypothesis follows.
Work derived from the piston expander can be expressed in terms of an electromagnetic pseudo-scalar gauge-invariantly coupled to the gauge field
Φ and the critical length exponent
v. The relationship is in agreement with the cosmological inflation model proposed by Ratra [
29]:
where the covariant vector
Fμν and contravariant gradient potential
Fμν combine to produce Lorentz invariance for the pseudo-scalar field when rotated on a hyperbolic manifold, ie. the electromagnetic field pseudo-scalar enables a non-additive energy contribution to enter the non-equilibrium system in the form of hyperbolic curvature.
In the quasi-micro-canonical ensemble [
2], the electromagnetic field pseudo-scalar is involved in the coupling mechanism but contributes no work in itself. It expresses the Berry curvature of the vacuum manifold whilst hosting the magnetic exchange pathways that facilitate energy transfers either to or from the vacuum manifold. The inner-product of the E and B fields remains the same viewed in all relativistic frames [
30] with the pseudo-scalar field remaining Lorentz invariant such that:
where
c is the speed of light.
Ginzburg-Landau theory states that the free energy of a superconductor near a phase transition can be expressed in terms of a complex order parameter field [
31]:
Then a complex rendering of the coupling energy term
e3vΦ maps to a complex wavefunction of the Berry phase:
where the quantity |
Ψ(
r)|
2 reflects the density of superconducting charge carriers; electrons for Type-II and the magnetic counterpart arising from gauge monopole charges for dual Type-I [
13].
Appendix A provides a summary of the Ginzburg-Landau theory of superconductors.
In the dual superconductor model of confinement [
11,
12], the Yang–Mills vacuum is based on the condensate of a magnetically charged Higgs field. In this situation, the critical correlation length
ξ also represents the coherence length
ξ’ of the magnetic monopole condensate [
14] which diverges to encompass total hyperbolic volume
V of the system at the superconducting phase transition [
31]. In this case the coherence length
ξ’ becomes exceptionally large under relativistic Lorentz rotation [
2].
ξ’ also gives the distance over which the dual superconductor can be represented by a wavefunction.
Since the coherence length
ξ’ and maximum value for the Ginzburg-Landau parameter
κ for the Type-I dual superconductor are known [
2], the London penetration depth
λ can be derived (see
Appendix A).
ξ’ and
λ are equal to the inverse Higgs mass
mH and inverse vector boson mass
mV, respectively [
13]. In normal metallic superconductors
λ is the distance within which an externally applied magnetic field disappears inside the superconductor. However, for the dual superconductor
λ represents a distance beyond the developing QCD flux tubes within which the magnetic current and electric field are expelled as a result of the dual Meissner effect.
So, mH determines the extent of QCD vacuum, which manifests in the embedding vacuum manifold volume V and mV determines the Gaussian hyperbolic curvature K of the embedding vacuum manifold.
Appendix A includes supporting quantitative analysis.
The complex form of the coupling energy term resembles a quantum mechanical wavefunction in which the energy spectrum is made entirely real and observable through dissipative structuring of water ice cages. Ψ0(r) corresponds to the emergence of the gauge field Φ at the Type-II to dual Type-I superconducting phase transition. Dissipation of either the scalar field or the critical correlation length ξ would represent a collapse in the wavefunction.
Application of de Moivre’s formula and isomorphic mapping of the complex field to rotational matrix form gives:
and similarly expressing electromagnetic duality as rotations in the 2-dimensional real plane:
Then the conjugate transpose of (8) is (9) and VV
H = 1 suggesting that
PV work of the piston expander is contingent upon the decomposition of a Hermitian unitary matrix A into two 2 x 2 non-Hermitian unitary matrices (ie. two complex matrices V and V
H containing both real and imaginary components such that V
H ≠ V) [
32]. The gauge field and the electromagnetic pseudo-scalar are thereby coupled through a marginal interaction.
Although this interpretation appears at odds with the expression for
PV work stated in (4), in fact any 2 x 2 complex symmetric matrix A can be eigendecomposed into a diagonal matrix D sandwiched between two complex unitary matrices, ie. VDV
H in this case. Minkowski spacetime vectors can be represented by 2 x 2 orthogonally diagonalizable matrices and incorporated into the extended physical VDV
H decomposition to reveal the coupling energy source:
These Hermitian matrices exhibit basic 3-dimensional rotation as well as 4-dimensional Lorentz transformation properties consistent with the relativistic length expansion and time contraction associated with the non-extensive element of
PV work, as revealed through the experimental results [
2]. Thus, the 2 x 2 unitary matrix A as a member of the
U(2) symmetry group is decomposed into factors identifiable as both Hermitian and non-Hermitian.
When represented in terms of gauge symmetry groups [
32], the
U(1) group of electromagnetism (via its mapping to
SO(2) in the 2-dimensional real plane) and the
SU(2) group of the complex order parameter
Ψ(
r), are in fact subgroups of the
U(2) group such that:
which describes a mapping to a Yang-Mills electroweak symmetry group [
33] where ℤ
2 represents the topology associated with the condensation of gauge monopoles [
34]. Formation of the
U(2) group is accompanied by critical behaviour and emergence of the gauge field
Φ as predicted by the Yang-Mills theory.
The dual superconductor model has several interpretations that require condensation of gauge monopoles, just as normal superconductivity results from the condensation of electric charges (or Cooper pairs – see
Appendix A) [
13,
14]. Theoretical frameworks for the condensation of gauge monopoles have been structured in terms of Abelian gauge-invariance (the
SU(2) gauge symmetry group) or non-Abelian gauge-invariance (the
SU(3) gauge symmetry group). Recent efforts [
14] have sought to extract the Abelian component responsible for gauge-independent quark confinement from non-Abelian gauge-invariance required for gauge monopole condensation without losing the essential characteristic of asymptotic freedom. From the experimental findings [
2] such a solution emerges out of an electroweak interaction that preserves asymptotic freedom, as described below.
In the vacuum of a dual superconductor, the dual Meissner effect compresses the chromoelectric flux between a quark and antiquark into a thin flux tube to form the hadronic string [
13,
35]. As the distance between quark and antiquark increases, the flux tube becomes longer whilst maintaining a minimal thickness. This geometry ensures that the energy increases linearly with length to create a linear confining potential between the quark and antiquark that bears a similarity to the linear oscillating Hamiltonian of the system. The flux tube determines the extent of QCD vacuum suppression, ie. positions where the colour-electric field is maximally expelled to leave a residual dual superconductivity [
36].
Yang-Mills theory requires the existence of both chromomagnetic monopole condensation (given by a coherence length) and the dual Meissner effect (given by a penetration depth) [
13,
14]. The force carrying gauge bosons of QCD are gluons which perform a similar role to photons in electromagnetism. Since the gluon field represents a local expulsion of the QCD vacuum, the absorption of physical gluon emissions into the QCD vacuum would tend to reduce local ‘space density’ and effective magnetic permeability
μ0. The net effect is to increase the hyperbolic curvature
K of the embedding manifold, ie. a quantum mechanical process manifests as ‘strong gravity’ [
37].
Berry Phase and Parity-Time (PT) Symmetry
The foundations of Berry phase physics lie in the adiabatic theorem of quantum mechanics [
38] which provides a formal description for a system coupled to a slowly changing environment. If the system Hamiltonian
H(
t) varies adiabatically and │
Ψ(
t)⟩ is an associated eigenstate then, following cyclic evolution of the environmental parameters where
H(
T) =
H(0), the state returns to itself but gains an additional phase factor [
38,
39]:
where
α represents the angular momentum of the wavefunction. For a dual superconductor, it originates from the exclusion of momentum resulting from chromomagnetic monopole condensation which is effectively stored in the electromagnetic field pseudo-scalar [
24].
The adiabatic theorem is based upon a single, non-degenerate eigenstate to which the system ‘clings’ as the environment is slowly changed [
40]. However, for the pressure-perturbed system being examined, asymptotic freedom constrains innumerable, degenerate and excited eigenstates to a singular value of total energy in the oscillating Hamiltonian function. External pressure perturbations applied to the crystal-fluid material see changes in kinetic/ internal energy mirrored by changes in negative energy potential such that total energy remains constant. When perturbations cease, the negative energy potential dissipates but internal energy becomes fixed close to the final resting value. So, in this case, a positive or negative perturbation of any duration is responsible for a single linear oscillation, or cycle, that is imposed upon a linear, sliding-scale of discrete values. Integration of the scalar potential ∇
Φ over a Hamiltonian cycle reveals the gradient energy term -½(∇
Φ)
2 that becomes observable in the
PV work extracted from the piston expander (see Equation (22) below).
The transient negative energy potential responsible for the phase factor
Ψ(
T) exists only for the duration of the pressure perturbation. Whilst the Hamiltonian remains constant under perturbation (as
Figure 2), it resolves to a different, stable value once the perturbation ceases. The final value of internal energy is then ‘propped’ and stabilized through dissipative structuring of water ice cages, subject to limited dielectric relaxation, as quantified by Stage 2-3 and Stage 4-1 (
Figure 2).
For acceleration (Stage 1-2) the effective radius decreases, and for deceleration (Stage 3-4) the effective radius increases. However, the resulting ‘hidden’ inertia is deemed not to be directly responsible for the Berry curvature term within the geometrical phase (7) since the externally-induced momentum manifests entirely in non-additivity of the hyperbolic curvature K. Instead, the Berry curvature is linked to the condensation of magnetic charges whereby the resulting exclusion of charge momentum manifests in the energy potential of the electromagnetic field pseudo-scalar (5). The Berry curvature is subsequently captured to be made real and observable in the variable hyperbolic volume V of the embedding vacuum manifold. Again, this hyperbolic volume is stabilized by the dissipative structuring of water ice cages within the crystal-fluid material so that the complex Berry phase is transformed into real work done.
For a classical thermodynamic system, changes in inertia ½mr2 represent changes in kinetic/ internal energy. However, since both internal energy and specific volume are highly constrained parameters within a false vacuum system, the energy of acceleration/ deceleration is prevented from manifesting in the crystal-fluid material. Thus, Pv work is limited to interactions with the walls of the vessel. For the synchronized U(2) symmetry group identified below, angular momentum is instead conserved in the acceleration/ deceleration of quarks that results in the emission/ absorption of gluons, ie. changes in negative energy potential. Gluons emitted by quarks are absorbed by the QCD vacuum manifold whilst the gluons absorbed by quarks emerge from the QCD manifold, thereby tending to effect local ‘space density’ and effective permeability μ0.
Pv is insignificant in comparison to
PV such that it represents the negative energy potential of the crystal-fluid material only. Therefore, for a constant Hamiltonian oscillator of constant mass
m, ½
r2 ∝ 1/
Pv, as described in Appendix B of [
2]. The average 1-dimensional radius
rx of the stable, pre-critical system is then found:
The Gaussian curvature
K for the 2-dimensional, hyperbolic surface of the pre-critical system (ie. with no topological defects) for the principal curvature relationship of
rx = -
ry, can then be determined:
or
Then, the average Gaussian radius of hyperbolic curvature (1/
K or
Rg) is given by:
Gaussian curvature
K has units of m
2 s
-2 that map directly to the vector boson mass
mV as the inverse of the penetration depth
λ (as described in
Appendix A). Through this mechanism, the negative energy potential of gluons is conserved through indirect hyperbolic curvature quantifiable by the non-equilibrium values of pressure
P and specific volume
v. Thereby, a quantum mechanical action is tuned thermodynamically under false vacuum conditions.
Decomposition of the complex gauge connection (e
iΦ)
3ν in Equations (8) and (9) suggests that complex Berry curvature is necessary for emergence of real coupling energy (1). It also determines the phase of electromagnetic duality, which in the extreme leads to dual superconducting behaviour, ie. condensation of magnetic charges resulting in the exclusion of magnetic current and the electric field. The cyclic evolution of the gauge connection results from the effective adiabatic property of the constrained false vacuum system (as revealed in the constant Hamiltonian oscillator of
Figure 2) to establish a novel form of the Berry phase [
38], one responsible for topological ordering in the dual Type-I superconductor [
10]. As with the conventional ground-state Berry phase, this ‘excited-states’ variant exposes the gauge structure in quantum mechanics [
15,
41].
In addition to describing the emergence of a gauge field
Φ, the gradient energy term -½(∇
Φ)
2 of the Lagrangian also maps to the complex order parameter field
Ψ(
r) in accordance with Ginzburg-Landau theory. The
PV work generated in the piston expander suggests that the associated quantum mechanical wavefunction is made real and observable, a phenomenon recently uncovered by Gu et al. [
15]. More precisely, the VDV
H decomposition reveals that the wavefunction becomes entirely real as the coupling energy is exposed through the diagonal matrix D in the VDV
H decomposition (10).
Since the system can be described through a combination of Hermitian and non-Hermitian matrices, it resembles a PT symmetric system [
16]. Such systems are characterized as not being isolated from the environment (ie. non-adiabatic) but subject to highly constrained interactions. This description is consistent with the false vacuum behaviour of the crystal-fluid material where both specific volume and internal energy are highly constrained. Energy and entropy gains and losses to the environment (including the embedding vacuum manifold in this case) are exactly balanced, ie. a renormalized, scale-invariant interaction between condensed matter and quantum wavefunction becomes evident in the constant energy Hamiltonian.
PT symmetry requires both space reflection and time reversal symmetries. The upside-down potential of the quartic term as identified by Bender [
16] is consistent with the marginal interaction and negative gradient energy term derived from experiment [
2]. However, the results presented here reveal the symmetry of Lorentz boosts, ie. symmetries in the expansion and contraction of both space and time, which may represent a more generalized form of PT symmetry.
Symmetry Synchronization and Conserved Quantities
Quark acceleration produces gluon emissions since a lower binding potential is necessary to maintain the momentum and energy of any given quark colour configuration [
42]. This results from a gluon recombination process whereby a quark and antiquark pair are annihilated. The emergence and absorption of physical gluons represents an exchange between the non-Abelian gauge symmetry of QCD and the Abelian gauge symmetry of the vacuum manifold, ie. an electroweak interaction.
The following non-Abelian Faddeev-Niemi decomposition is considered [
43]:
This decomposition is a restricted one since splitting and recombining gluons in
SU(3) represents a limited interaction with a
U(2) spacetime manifold rather than full symmetry breaking to
SU(2). The requirement for a Higgs-type scalar field is satisfied by the emergent gauge field
Φ [
14]. So:
Asymptotic freedom is thereby maintained through the dominant
SU(3) group. Again,
U(2) appears as an electroweak symmetry group [
33] with ℤ
2 representing a topology consistent with the condensation of gauge monopoles [
34].
A U(2) gauge symmetry that provides for the condensation of gauge monopoles has so far been identified in both the condensed matter system and the underlying QCD particle physics. However, it is also possible to determine a U(2) gauge symmetry for the vacuum manifold of local spacetime through which hyperbolic curvature and scalar field potential are effected. That is, where the splitting and recombining of force-carrying gluons are associated with fictitious forces in non-inertial reference frames.
The Lorentz group
SO(4) provides for the conservation of energy and angular momentum in 4-dimensions (ℝ
4) through two continuous symmetries; rotations in 3-dimensional Euclidean space and Lorentz boosts which influence both space and time [
44]. The 4 x 4 orthogonal matrix representation of the metric tensor can also be cast in terms of a 2 x 2 unitary matrix operating on a complex 2-component spinor. The complete unitary 2 x 2 transformation matrix for spinor rotations and boosts can be expressed as:
or
where
θ is the Lorentz rotation angle,
σ is the Pauli spin matrix, and ϕ is the angle associated with the Lorentz boost (or rapidity) [
45]. Equation (19) represents a ‘right-handed’ spinor
ϕR and (20) represents a ‘left-handed’ spinor
ϕL, ie. the Weyl spinors. Later insights by Dirac led to the concept of the bispinor which, unlike (19) and (20), preserves parity of the wavefunction under the sign reversal operation
Ψ(
x,
t) →
Ψ(-
x,
t) thereby maintaining a positive gauge field and positive energy (whilst also predicting the existence of antimatter). However, retaining the 2 x 2 unitary matrix whilst acknowledging parity preservation requirements produces the following spacetime group representation [
46]:
The symmetry group decompositions in (11), (18) and (21) are then amalgamated to describe a consolidated ‘symmetry synchronization’ that establishes common scale- and gauge-invariance in
U(2), as shown schematically in
Figure 4:
When a symmetry is broken, a corresponding order parameter that diminishes to zero can often be identified. However, in this case the complex order parameter Ψ(r) emerges where symmetry is synchronized.
Both energy and angular momentum are conserved within the common U(2) group to reveal the time and space symmetries of a Lorentz boost in agreement with Noether’s theorem (see below). Since there is a gluon field for each colour charge, it follows that each gluon field can be composed of a time-like component and three space-like components. These components relate to the electric potential and the magnetic potential, respectively, and will interact with the vacuum manifold to determine the values of effective permittivity ε0 and effective permeability μ0.
Variations in effective
μ0 require that a spontaneous magnetic flux M
s, with associated spontaneous magnetic field H
s, emerges to conserve magnetic charge. Fractionalized magnetic charges arising from the geometrically frustrated crystal-fluid material can be interpreted as condensing into a gauge monopole topology that excludes magnetic current to provide magnetic exchange pathways. The correlation length
ξ of the magnetic monopole condensate produces divergent critical behaviour that is shown to have a distinctive universality class of critical exponents. The gauge monopole topological defects act as both convergent sinks (under acceleration) and divergent sources (under deceleration) of the magnetic flux M
s [
10]. The nature of these defects is speculated in
Appendix A.
Similar principles apply to variations in vacuum energy determined by the local ‘space density’ (which determines the embedding manifold curvature). Conservation of energy requires that negative
PV work is performed under false vacuum acceleration (energy is transferred to the vacuum manifold) whilst positive
PV work is performed under deceleration (energy is transferred from the vacuum manifold). Work is related to the gauge/ scalar field
Φ as follows [
2]:
The right-side of Equation (22) represents the gradient energy term of the Lagrangian function resulting from the scalar potential ∇
Φ developed across the gauge monopole topology to give the integral of the scalar potential ∇
Φ. The Lagrangian action of the left-side, ie. mechanical work, is also related to the critical response function revealed in the coupling relationship (1) to confirm energy renormalization in the synchronized
U(2) group complex parameter field
Ψ(
r). That is, energy equivalence between the long-range dissipative structuring of water ice cages and the short-range confinement mechanisms of sub-atomic particles, as illustrated in
Figure 4. This outcome aligns with Anderson’s speculative prediction [
47]:
‘Physics in the 20th century solved the problems of constructing hierarchical levels which obeyed clear-cut generalizations within themselves […]. In the 21st century one revolution which can take place is the construction of generalizations which jump and jumble the hierarchies, or generalizations which allow scale-free or scale transcending phenomena. The paradigm for the first is broken symmetry, for the second self-organized criticality.’
With
U(2) scale- and gauge-invariance spanning the asymptotically-free behaviour of both the macro-scale dual superconducting system and the quark-gluon system via interactions with the embedding vacuum manifold, a physical correspondence between non-equilibrium thermodynamics and quantum mechanics is established. Since the superconducting phase transition is represented by Ginzburg-Landau theory (ie. gauge-invariant coupling of a scalar field to the Yang-Mills action is predicted) it seems reasonable to link the gradient energy gap of
Figure 3 to the mass gap problem in QCD.
Gapped and Gapless Topologies
The results in
Figure 3 show emergence of the gauge field
Φ as a gap between Type-II superconductivity on the left and dual Type-I superconductivity on the right. This represents a transition between the gapped state of the magnetically ordered Type-II superconductor and gapless state of the topologically ordered dual Type-I superconductor. At this point, the gauge monopole charges condense and the electric field E is excluded to be confined on the surface of the system (prior to the emergence of topological defects that penetrate the magnetic condensate). That is, a gapless surface is established so that the Berry phase manifests as a non-trivial topological insulator [
38].
The gapless surface may be protected from external perturbations tending to re-open the gap through non-Abelian topology, as represented by the ℤ
2 Chern number in the symmetry group decompositions of (11) and (18). In a review of topological superconductors [
48], Sato and Ando explore the connection between ℤ
2 and time reversal symmetry that is consistent with the symmetry of Lorentz boosts described above. The ℤ
2 Chern number can be interpreted as a
U(2) group that fibres over a circle as a 3-sphere bundle, ie. a Hopf fibration results [
49].
Typically, a topological insulator is characterized by a non-robust, non-degenerate ground-state in which energy bands coincide and exceptional, or ‘diabolical’, points occur. However, the Berry phase variant identified above displays the following features: asymptotic behaviour (robustness against perturbations); a critical correlation length
ξ (long-range entanglement); conformal geometry (describable through quantum field theory); and degeneracy in non-trivial topology on a hyperbolic manifold [
49]. Thus, the system also appears to be topologically ordered and so describable by an effective, low-energy topological quantum field theory (TQFT) in which many-body states would typically have topological ground-state degeneracy [
50]. In TQFT the critical correlation length
ξ is topologically invariant and therefore insensitive to the geometry of the embedding manifold, ie. the critical exponents within the universality class remain constant under Lorentz boosts.
Within the research field of topological phases of matter, as investigated to date, all the topologically ordered states realized experimentally or investigated theoretically are established through strong electron-electron interactions. The coinciding valence bands of gapless ‘diabolical’ points allow for degenerate electron movements between the bands. In a crystal structure, the electronic band structures are described by Bloch’s theorem as expressed by:
where
Ψ is the wavefunction,
r is position,
u is a periodic function, and
k is the crystal momentum vector.
However, the original formulation of the Berry phase was not specifically related to Bloch electrons. Instead, it was based on the general idea that quantum adiabatic transport of particles in slowly varying fields (eg. electric, magnetic, or strain) could in principle modify the wavefunction by terms other than just the dynamical phase. So, Equation (23) is seen to map to the experimentally derived Equation (7) where variable electromagnetic duality of the pseudo-scalar field gives the periodic function
u(
r) and (
eiΦ(r))
3v represents the angular momentum of magnetic charges that becomes effectively stored in the pseudo-scalar field as the magnetic monopole condensate forms [
26].
Since the degeneracy associated with the crystal-fluid material occurs in metastable excited-states (with non-zero temperatures), the gapless degeneracy of the crystal-fluid material cannot be attributed to Bloch electrons. So, whilst the gapless surface of the dual superconductor is protected through ℤ2 topology, an additional mechanism is necessary to supress excited-state fluctuations such that Bloch-wave behaviour can emerge.
A potential solution is presented in
Figure 4 where the synchronized continuous symmetry group
U(2) leads to descriptions of asymptotic freedom in both quantum and condensed matter systems. As conservation of angular momentum extends into the microscopic quantum realm, so confinement mechanisms extend out into the macroscopic condensed matter of the dual superconductor under a renormalized Noether symmetry. The non-Abelian
SU(3) group of QCD remains dominant so that excited-state fluctuations due to changes in momentum are suppressed through confinement mechanisms and the constant Hamiltonian function is preserved. Topological defects may represent the penetration of the dual superconductor by the excluded electric field in the form of chromoelectric flux tubes, so enabling the formation of quark-antiquark pairs together with an inherent confinement mechanism.
Acceleration and deceleration of the crystal-fluid material thereby become confined interactions responsible for the splitting and recombining of gluons. Gluons are either absorbed by or emerge from the QCD vacuum manifold. The Higgs-like gauge field
Φ also emerges in the transition from gapped Type-II superconductivity to gapless dual Type-I superconductivity to establish a reciprocal gap in the gradient energy, as revealed in
Figure 3. For the gapless state, the spin-1 vector gluons are the force-carrying
SU(3) gauge bosons/ quasiparticles that generate the Bloch-wave description in the magnetic condensate [
51]. Gluon interactions with the embedding QCD vacuum manifold enable fictitious forces to emerge in non-inertial reference frames that lead to
PV work in the piston expander.
Whilst the massless gluons either emitted or absorbed by accelerating or decelerating quarks act as gauge particles, gauge invariance is only established where the associated gauge field can emerge under U(2) ‘symmetry synchronization’ above a critical correlation length ξ in the low-energy system. That is, gauge monopole charges are required to condense so that the scalar field potential induces a flow of magnetic charge such that ∇. Bs = 0, where Bs is the spontaneous magnetic flux density.
By extension, the lower bound of the QCD mass gap can be attributed to a symmetry-breaking of
U(2) and the evaporation of a gauge monopole condensate from below in the low-energy, infrared limit. In this event, the conservation pathway between the condensed matter and quantum systems no longer exists since the systems are effectively isolated except for weak residual gravitational interactions, ie. the systems are decoupled. With this spontaneous breaking of
U(2) gauge symmetry into isolated sub-groups, and the collapse of any critical magnetic correlation length
ξ, there is no mechanism through which non-equilibrium angular momentum can be conserved through gluon interactions and the corresponding curvature imposed on the embedding manifold, ie. conservation of energy and momentum becomes limited to the individual symmetry groups and the condensed matter system becomes describable by classical thermodynamics and discontinuous (first order) phase transitions. Thus, in this interpretation of broken gauge symmetry, a strong gravitational interaction [
37] is replaced by a far weaker one essentially limited to gluon interactions arising from quantum fluctuations and other irrelevant interactions. In other words, it becomes non-renormalizable.
Cosmological Analogy
The same principle may be applied to the high-energy bound of the QCD mass gap from above. Guth’s model of cosmological inflation [
52,
53] is also founded upon false vacuum and negative pressure conditions. The associated deceleration of inflationary expansion within a false vacuum would result in gluon splitting where conservation of energy and momentum are mediated by a synchronized order parameter field/ symmetry group together with the corresponding cosmic monopole condensate necessary to facilitate a critical coherence length and penetration depth. The splitting of gluons and the emergence of quark-antiquark pairs act to increase the strong interaction through confinement mechanisms.
The non-extensive inflationary volume expansion model is [
2]:
where
Vr is the reduced volume │(
V –
Vc)/
Vc│ at the critical temperature
Tc with a correlation length exponent
v in 3-dimensions. Critical volume
Vc and critical temperature
Tc are not absolute values but rather ‘rolling’ dynamical values determined by structural anisotropy and dissipative false vacuum restructuring of elementary particles under non-equilibrium conditions. At the collapse of critical behaviour,
Vr = 1 and non-extensive inflationary volume expansion ceases.
The critical length exponent
v represents the dimensionless group parameter of rapidity such that its emergence on a Lorentz hyperbolic manifold (ie. a Lorentz boost) is associated with a relativistic length expansion to describe non-extensive inflationary volume expansion [
2]. Deceleration increases the effective radius of structural elements under false vacuum conditions such that angular momentum is removed from the sub-atomic quarks. The counteracting emergence of quark- antiquark pairs
qq̄ plus gluons from the QCD vacuum, the associated gluon splitting, together with subsequent integration of the emergent particles into complex binding arrangements, establishes the colour and quark confinement mechanisms. These tend to increase local ‘space density’ and effective magnetic permeability
μ0. The resultant effect sees hyperbolic curvature
K of the embedding vacuum manifold reduce leading to inflationary expansion work. The process is shown as an analogue of the experimental findings in
Figure 5.
At the collapse of critical behaviour, the synchronized gauge symmetry is broken, the cosmic monopole condensate with associated topology evaporates, and critical correlation length is destroyed. Absorption of the gauge field into the QCD colour field attributes mass to the emergent quark and antiquark pairs qq̄ of the confinement process under a Higgs-like mechanism associated with an electroweak interaction, whilst volume and internal energy are ‘propped’ and stabilized by reorganizing dissipative structural elements. Again, under such an interpretation, classical thermodynamics essentially separates from quantum mechanics to leave only weak residual gravitational interactions, ie. the strong interaction becomes short-ranged. The energy and mass of the strong interaction are thereby effectively fixed at the point where the common symmetry group associated with cosmological inflation is broken, the cosmic monopole condensate evaporates, and the critical correlation length is destroyed.