Energy Renormalization in a Berry Geometrical Phase: Low-Energy Perturbations of the Strong Interaction and the QCD Mass Gap

Introduction

Experimental Evidence and Background Material

Experimental Setup

The temperature and pressure of the crystal-fluid are measured at five-second intervals with sensors that have direct contact with the crystal-fluid and recorded by a PLC/ PC monitoring system. All values for energy and thermodynamic potentials are derived from the pressure and temperature measurements by the NIST REFPROP program/database [17]. The calculations are in accordance with GERG-2008 modified by the Kunz and Wagner Model 0 (KW0) [18]. The piston expander is completely immersed in a heat bath with a temperature of 270K, approx. The schematic arrangement provided in Figure 1 is reproduced from [1].

Figure 1. Schematic arrangement of the experimental apparatus in which a negative pressure material is formulated and manipulated. Naively, dissipative structuring of the crystal-fluid material is controlled with a view to establishing a power cycle in the piston expander through non-equilibrium, non-extensive volume displacements.

Figure 1. Schematic arrangement of the experimental apparatus in which a negative pressure material is formulated and manipulated. Naively, dissipative structuring of the crystal-fluid material is controlled with a view to establishing a power cycle in the piston expander through non-equilibrium, non-extensive volume displacements.

A negative pressure fluid is established by the Berthelot method [19]. Approx. 3.5 grams of crystal-fluid are transferred into a previously evacuated stainless-steel sample vessel (50ml). A low-energy, negative pressure regime results in the formation of water ice cages hosting methane molecules. The sample vessel is completely immersed in a relatively large heat bath (70 litres) where the temperature of the bath is controlled with an electric element and a refrigeration dip cooler. Once the desired temperature is obtained, the sample is released into the fluid-side of the 0.5 litre retracted piston expander, also completely immersed in the heat bath, which displaces the piston vertically upwards to the fully-extended position. The gas-side of the piston is open to atmospheric pressure during the extension. This action reduces the energy of the system further and is intended to transfer the guest methane molecules from the host water ice cages to similar structures within the inhibitor solvent. An increase in mass of 0.3 kg, approx. is immediately recorded against the first piston extension. Negative and positive piston displacements are then induced through pressurized nitrogen perturbations to produce negative and positive work outputs where displacement ratios are 1:100 and 100:1, approx.

The Phenomena

From completion of the Berthelot mixing process through subsequent positive and negative displacements of the piston expander, REFPROP determines that the crystal-fluid material remains in a subcooled liquid phase, as recorded in Appendix A. It is both astonishing and remarkable that 3.5 grams of material does not transition to a vapour or gas, nor produce any methane outgassing, when contained within the initial sample volume of 50 ml. More remarkable still is that an additional work-generating positive piston displacement of 0.5 litre also has no effect upon the integrity of the subcooled liquid phase. Notwithstanding, a conventional interpretation in terms of fluid mechanics would locate all the crystal-fluid material in the lowest section of stainless-steel tubing connecting the sample vessel to the piston expander (excluding any capillary action) due to ordinary gravity and the generation of work would be inconceivable. Additionally, the mass increase of 0.3 kg, approx. developed during the first piston extension is maintained throughout subsequent piston displacement actions. This perplexing and counterintuitive outcome is examined in more detail below together with supporting mathematical expressions.

In addition to the temperature and pressure measurements, only the piston position and mass of the material components are required to calculate all the thermodynamic properties, critical exponents and scaling relations shown in Appendix A. Whilst validity of the REFPROP calculations may be reasonably challenged, it has been demonstrated successfully that the program/ database is very sensitive to outgassing and re-absorption events associated with phase transitions in similar materials when performing quasi-thermodynamic cycles [5]. In such circumstances methane outgassing accompanies the formation of low-energy, guest-free water ice cages and is consistent with the fluctuation-dissipation theorem. With these phase-change processes, long-range interactions are also established whereby non-additivity in the fundamental thermodynamic relation is revealed.

Figure 2 is reproduced (with some additional annotation) from the recent experimental results reported in [1] where Points 1-4 identify particular stages of the work cycle in a low-energy system; Stage 1-2 corresponds to negative displacement of the 0.5 litre piston expander and Stage 3-4 corresponds to positive displacement.

Figure 2. Excess internal energy potential resulting from excess thermodynamic potentials during external pressure perturbations. All values for energy, work and thermodynamic potentials are derived by the NIST REFPROP program/database based only upon temperature, pressure and mass measurements together with the piston position. The calculations are in accordance with GERG-2008 modified by the Kunz and Wagner Model 0 (KW0). Each piston displacement is associated with approximately 203 kJ of energy, 8.5 kW of power and 2.9 MN of force.

Figure 2. Excess internal energy potential resulting from excess thermodynamic potentials during external pressure perturbations. All values for energy, work and thermodynamic potentials are derived by the NIST REFPROP program/database based only upon temperature, pressure and mass measurements together with the piston position. The calculations are in accordance with GERG-2008 modified by the Kunz and Wagner Model 0 (KW0). Each piston displacement is associated with approximately 203 kJ of energy, 8.5 kW of power and 2.9 MN of force.

Changes in negative potential energy and internal energy vary in a 1:1 relationship after discounting the Pv work term associated with the walls of the vessel. Thus, a linear oscillator or constant energy Hamiltonian function is determined. The associated affine non-concave entropy function [1,2] represents local stability conditions achieved through dynamically responsive inhomogeneities, i.e., the complex reorganization of dissipative structures composed of water ice cages, such that energy cannot be minimized and entropy cannot be maximized [2]. Specific volume and internal energy are shown to be highly constrained intensive parameters.

Phase Transitions and Collective Phenomena

Loss of homogeneity is characteristic of phase transitions that give rise to critical phenomena. The discovery of a distinctive universality class of critical exponents that satisfy the scaling laws of Fisher, Rushbrooke, Widom and Josephson [21,22] reveals spontaneous magnetic and superconducting properties for the crystal-fluid material investigated. A phase transition from Type-II superconductivity to dual Type-I superconductivity is identified through the Ginzburg-Landau parameter κ where the spontaneous magnetic field H_s transitions from negative to positive (Appendix C) and a complex order parameter field Ψ(r) emerges. Spontaneous magnetism M_s can either increase or decrease dependent upon the change in direction of hyperbolic curvature K. Positive piston displacement results from a reduction in K to give increasing values of M_s. Negative piston displacement results from an increase in K to give reducing values of M_s.

Derivation of the Gaussian hyperbolic curvature K also gives the Gaussian radius R_g as its inverse, as shown below. This enables the hyperbolic surface area of a hollow, walled sphere having effective radius R to be determined (A = 4πsinh²(R/2)) [1]. The hyperbolic surface area maps almost exactly to the negative inverse of the gradient energy, where topological defects are introduced, in an apparent expression of holographic duality [35]. The external pressure perturbations may be interrupted at any point such that both the swept volume V and the hyperbolic curvature K become fixed and stable. This suggests that the non-equilibrium gradient energy is captured and confined within the hydrogen bonding interactions of the water ice cages and inhibitor solvent as the system relaxes into a non-critical, stable state (refer to equations (1) and (2)).

Inequalities in the associated Maxwell relations together with calculations of the hyperbolic geometry reveal non-additivity and non-extensivity in the fundamental thermodynamic relation. Additivity can be restored through hyperbolic curvature (i.e., surface area A) whilst extensivity can be restored through gradient energy/ coupling energy (i.e., volume V) [22]. The coupling energy is related to the critical correlation length exponent ν in 3-dimensions combined with values of a scalar field Φ as derived from the gradient energy term -½(∇Φ)² of the Lagrangian such that:

$${\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}\propto \left(e^{\Phi }\right)}^{3v}$$

(5)

The critical correlation length ξ associated with the universality class is linked to volume V through Lorentz boosts, i.e., relativistic velocity and the reference frames associated with acceleration and deceleration [1]. The critical correlation length ξ is revealed to be a Lorentz length expansion; a relativistic phenomenon coinciding with the formation of a magnetic condensate [12]. The relativistic time contraction conjugate to the critical correlation length is also revealed in Figure 3 where the y-axis representing scalar field values has units of s^-1, i.e., the reciprocal of time.

Figure 3. Effect of the critical correlation length ξ on the magnitude of the scalar field Φ. A gradient energy term -½(∇Φ)², as derived from the non-equilibrium Lagrangian function, is equivalent in magnitude to the coupling energy. From this, scalar field Φ values associated with the critical response are determined. Changes in negative gradient energy and positive coupling energy are both equivalent in magnitude to PV work done by the piston expander which can be either negative or positive.

Figure 3. Effect of the critical correlation length ξ on the magnitude of the scalar field Φ. A gradient energy term -½(∇Φ)², as derived from the non-equilibrium Lagrangian function, is equivalent in magnitude to the coupling energy. From this, scalar field Φ values associated with the critical response are determined. Changes in negative gradient energy and positive coupling energy are both equivalent in magnitude to PV work done by the piston expander which can be either negative or positive.

Again, the correlation length ξ is associated with a superconducting phase transition and related to a Lorentz boost in 3-dimensional space (Appendix B). Its value reveals the presence of self-organized criticality responsible for a ‘rolling’ critical response in accordance with:

$$\xi \mathrm{}~\mathrm{}{│T- T_c│}^{-v}$$

(6)

where T is the system temperature and T_c is the critical temperature. Both temperatures are dynamic under external pressure perturbation to reveal sustained anisotropy in the water ice cage structures in either direction. The magnetic correlation length ξ corresponds to a gap in gradient energy that can be described by a relativistic length expansion having an associated conjugate time contraction [1].

The gradient energy term -½(∇Φ)² equates with either positive or negative PV work (i.e., it is described by the least action principle of the Lagrangian) whilst the hyperbolic surface area is a function of the Gaussian radius of hyperbolic curvature R_g. Both properties are calculated from experimental results (Appendix A). In order for the surface of the system to be fully expressed in the gradient energy, it is necessary to introduce topological defects at the superconducting phase transition where the spontaneous magnetic field H_s moves from negative to positive through zero.

Analysis and Discussion—New Insights

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 [56]. This results from a gluon recombination process whereby a quark and antiquark pair are annihilated. The emergence and absorption of gluons signifies a mediated exchange between the non-Abelian gauge symmetry of QCD and the Abelian gauge symmetry of the vacuum manifold, i.e., an electroweak interaction. In other words, the parameter space of the quantum mechanics is the spacetime manifold of an Abelian gauge theory such that the closed cycle of the Berry phase is formally identical to a Wilson loop observable [57], as demonstrated below.

The following non-Abelian Faddeev-Niemi decomposition is considered [58]:

$$SU\left(N+1\right)\to {\displaystyle \frac{SU\left(N\right)\otimes U\left(1\right)}{{\mathbb{Z}}_N}} ~ U\left(N\right)$$

(23)

This decomposition is a restricted one since splitting and recombining gluons in SU(3) represents a mediated 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 [12]. So:

$$SU\left(3\right)\to {\displaystyle \frac{SU\left(2\right)\otimes U\left(1\right)}{{\mathbb{Z}}_2}} ~ U\left(2\right) ~ {\displaystyle \frac{SU\left(2\right)\otimes U\left(1\right)}{{\mathbb{Z}}_2}} \to SU\left(3\right)$$

(24)

Asymptotic freedom is thereby maintained through the dominant SU(3) subgroup. Again, U(2) appears as an electroweak symmetry group [46] with ℤ₂ representing an Abelian topology consistent with gauge monopole condensation [47]. For Kondo et al. [12], U(2) is the maximal stability group determined by the fundamental quark group SU(3) where quark confinement is the non-Abelian micro-scale dual superconductivity associated with chromomagnetic monopoles. In the current exposition, this non-Abelian micro-scale dual superconductivity is subsumed into an Abelian gauge monopole condensate, as identified above, under the common U(2) symmetry of macro-scale dual superconductivity.

A U(2) gauge symmetry that provides for a gauge monopole condensate 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 is enacted. That is, where the splitting and recombining of massless, force-carrying gluons are associated with fictitious forces in non-inertial reference frames, as quantified by the effective vector boson mass (0.22 m²s^-2 ≤ m_V ≤ 0.46 m²s^-2).

The Lorentz group SO(4) provides for the conservation of energy and angular momentum in 4-dimensions (ℝ⁴) through two continuous symmetries; rotations in 3-dimensional Euclidean space and Lorentz boosts which influence both space and time [59]. 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:

$$U= e^{{\displaystyle \frac{1}{2}}i\sigma .\theta - {\displaystyle \frac{1}{2}}\sigma .\mathrm{\phi }}$$

(25)

or

$$U=e^{{\displaystyle \frac{1}{2}}i\sigma .\theta +{\displaystyle \frac{1}{2}}\sigma .\mathrm{\phi }}$$

(26)

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) [1,60]. Equation (25) represents a ‘right-handed’ spinor ϕ_R and (26) represents a ‘left-handed’ spinor ϕ_L, i.e., the Weyl spinors.

For bidirectional Lorentz boost transformations having no rotational components, the Weyl spinors simplify to ϕ_R = e^-σϕ/2 and ϕ_L = e^σϕ/2, which is the half angle formulation given in equations (14) and (15) above. It is observed that the critical correlation length ξ is equal to the average value of the combined rapidity angle exponentials with ±φ. A change in direction of the scalar potential ∇Φ is consistent with a reversal in direction of the rapidity angle exponents such that:

$$\xi ={\displaystyle \frac{e^{\pm 1.779}}{2}}+ {\displaystyle \frac{e^{\mp 1.779}}{2}}=3.05$$

(27)

Later insights by Dirac led to the concept of the 4-component spinor which, unlike (25) and (26), preserves parity of the wavefunction under the sign reversal operation Ψ(x,t) → Ψ(-x,t), thereby accommodating changes in scalar potential direction explicitly whilst maintaining a positive gauge field plus positive energy [45] (and also predicting the existence of antimatter). Then, retaining the 2 x 2 unitary matrix whilst acknowledging parity preservation requirements produces the following spacetime group representation [59,61]:

$$SO\left(4\right)\to SU\left(2\right)\oplus SU\left(2\right)$$

(28)

The symmetry group decompositions in (16), (24) and (28) 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:

Figure 4. Decomposition of symmetry groups to establish common U(2) scale- and gauge-invariance. This model of invariance provides a mechanism through which the pressure-induced ‘rolling’ critical response results in PV work that is either positive or negative. The complex reorganization of water ice cages produces variable inertia which, through the conservation of angular momentum, is responsible for either an acceleration or deceleration of quarks. In the case of acceleration, this leads to the emission of gluons that are indirectly absorbed into the QCD vacuum manifold. A corresponding tendency to reduce local ‘space density’ and effective magnetic permeability μ₀ manifests as increased hyperbolic curvature K.

Figure 4. Decomposition of symmetry groups to establish common U(2) scale- and gauge-invariance. This model of invariance provides a mechanism through which the pressure-induced ‘rolling’ critical response results in PV work that is either positive or negative. The complex reorganization of water ice cages produces variable inertia which, through the conservation of angular momentum, is responsible for either an acceleration or deceleration of quarks. In the case of acceleration, this leads to the emission of gluons that are indirectly absorbed into the QCD vacuum manifold. A corresponding tendency to reduce local ‘space density’ and effective magnetic permeability μ₀ manifests as increased hyperbolic curvature K.

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). There is a gluon field for each of the eight colour charges and each gluon field is 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 indirectly with the vacuum manifold to determine the values of effective permittivity ε₀ and effective permeability μ₀. Additionally, the chromoelectric and chromomagnetic components correspond to the temporal and spatial Wilson loops described below, where the Wilson loop represents an order parameter for quark confinement [4].

Variations in effective μ₀ require that a spontaneous magnetism 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 appear confined into magnetic entities through lepton decay. These electroweak magnetic monopoles are interpreted as condensing into a gauge monopole topology that excludes magnetic current to leave magnetic exchange pathways available for the spontaneous magnetism M_s. Whilst magnetic exchange interactions are conventionally established through direct and super-exchange mechanisms between metal centres or metal centres and various ligands, exchange can also occur via intermolecular hydrogen bonding interactions [62,63]. Changes in bond lengths and angles along the exchange pathway affect the hopping integrals between magnetic centres, thereby altering the magnetic exchange effect.

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 magnetism M_s [8]. The nature of these defects is speculated in Appendices C-E.

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 [1]:

$$\int P\mathrm{d}\mathit{V}=-{{\displaystyle \frac{1}{2}}\left(\nabla \Phi \right)}^2$$

(29)

The right-side of equation (29) represents the gradient energy term of the Lagrangian function resulting from the scalar potential ∇Φ developed across the gauge monopole topology which is integrated. The Lagrangian action of the left-side, i.e., 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 [64]:

‘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 micro-scale quark-gluon system via electroweak 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 (i.e., 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.

Conclusion

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