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We use bold face to indicate that some indices of the corresponding mathematical objects are suppressed. Consider the

1,1 Ψ1 H = Ψ = Ψ1,2 H (2.1)   Ψ2  ∈ 

consisting of vectors with two components, hencefor th called . The scalar product of two biquaternions Ψ, Φ H1,1 is defined by ∈ †a b Ψ Φ Ψ ΓabΦ . (2.2) · ≡ 2 The symbol stands for hermitian conjugation and Γab (a, b, ... =1, 2or1, 2, 3, 4, as needed) for a hermitian matrix with† quaternion entries and signature (+, ). The isometry group of H1,1 is Sp(1,1), consisting of all transformations which preserve the norm and products− of biquaternions. Using that Sp(1,1) is the minimal compact cover of O(1,4) (i.e., O(1,4) Sp(1,1)/Z2), we introduce a s Aa ∼ special Clifford algebra, l 1,4, such that its elements Γ b satisfy the following relations C ΓAΓB ΓBΓA = 2iσAB, (2.3) Aa Bb −Ba Ab − AB a Γ bΓ c +Γ bΓ c = 2η 1 c, (2.4) Γ0Γ1Γ2Γ3Γ5 = i1, (2.5)

where i is the complex , σAB is the so(1,4) generator of the rotation or boost in the plane (A, B) and 1 is the unit matrix. We further request

† † † Γ = Γ0, Γ = Γj, Γ = Γ5. (2.6) 0 − j 5 We note that the Γ matrices are not determined by Eqs. (2.3)-(2.6), since a conjugacy (or similarity) transfor- mation with a unitary matrix U (i.e., U † = U −1), ΓA U −1ΓAU, yields another set of matrices satisfying Eqs. (2.3)-(2.6). Two representations of the Γ matrices→ are provided in Appendix A. The cover relationship between Sp(1,1) and O(1,4) can be made explicit using the Γ matrices. For every S Sp(1,1), there exists Λ SO(1,4) such that ∈ ∈ −1 B S ΓAS =ΛA ΓB. (2.7)

In contrast, given Λ O(1,4), S Sp(1,1) is determined up to a sign. Still, SO(1,4) and Sp(1,1) share ∈ ∈ AB the same . An infinitesimal Sp(1,1) transformation, S = 1 iσABw /4, where wAB is an − B infinitesimal antisymmetric form, corresponds to an infinitesimal SO(1,4) transformation, written as ΛA = B B ηA + wA . The condition that an infinitesimal Sp(1,1) transformation leaves Ψ Φ invariant yields · Γσαβ σαβΓ = 0, (2.8) − Γσα0 + σα0Γ = 0, (2.9)

where α, β =1, 2, 3, 5. Γ is a solution of Eqs. (2.8) and (2.9) only if it is a linear combination (with complex 0 1 2 3 5 3 s coefficients) of Γ and Γ Γ Γ Γ . This concludes the construction of l 1,4. By convention, we use Γ and C † 2 its inverse to lower and raise the a, b, ... indices. We also note the identity ΓΓ = ΓAΓ and that (Γ) 1. A − ∝ 3 On 5D space-time and 5D null propagation

We discuss a particular class of sufficiently smooth 5D Lorentzian manifolds, equipped with a space-like Killing field, subject to the principle of general covariance. We denote such a manifold by and use M 2The indices a,b,... run over the of the Γ matrix. In turn, this depends on the algebraic representation of the quaternion units. Hamilton’s representation requires a,b,... = 1, 2. The representation based on the requires a,b,... = 1, 2, 3, 4. 3 s AB In this context, Cl 1,4 is naturally generated by the hermitian metric Γ and the Sp(1,1) generators, −iσ .

2 indices M,N, =0, 1, 2, 3, 5 for the global 5D geometry, while A, B, =0, 1, 2, 3, 5 stand for the local 5D geometry (i.e.,··· tangent spaces). Furthermore, we use the indices µ, ν,··· = 0, 1, 2, 3 for foliated curved 4D metrics and the indices α, β, =0, 1, 2, 3 for foliated flat 4D metrics. ··· ··· 5 We denote the 5D metric by hMN , choose the fifth coordinate x along the Killing field and foliate the 5 5D space-time along x . We denote the foliated 4D metric by gµν and the shift of the foliation by Nµ. Since we are only interested in 5D null propagation, we set the lapse of the foliation to 1 by a conformal transformation. The transformed 5D metric is denoted by h˜MN

µν µ ˜MN g˜ N ˜ g˜µν + NµNν Nµ h = ν ρ , hMN = − , (3.10)  N N Nρ +1   Nν 1  − whereg ˜µν stands for the transformed 4D metric. A To describe particle propagation, we choose a local 5D Lorentzian frame e M , called f¨unfbein, at each point of the 5D manifold

A B M N h˜MN = e M e N ηAB, ηAB = e Ae B h˜MN . (3.11)

s We further provide l 1,4 structure to the tangent space p at each point p of the manifold. That is, C A T A the f¨unfbein components of a tangent vector x at p yield an element x = x ΓA of a locally defined s 4 B B A s l 1,4 algebra. An element of p transforms with SO(1,4) (i.e., x ΛAx ), while an element of l 1,4 transformsC with Sp(1,1) (i.e., x T S−1xS). The biquaternion metric →Γ may change over the 5D space-timeC → and is considered on equal footing with the f¨unfbein metric, ηAB. In particular, we wish to describe the experience of uncharged local observers, who perceive geometrically the first four space-time dimensions, but not the fifth. We choose the following non-orthogonal f¨unfbein to reflect this

α µ A f µ 0 M f α 0 e M = , e A = µ , (3.12)  Nµ 1   Nµf α 1  − α where 0 denotes here a 4 1 matrix with null entries, and f µ is a local Lorentzian frame (vierbein) at each × point of the 4D space-time with the metricg ˜µν . We may further ask that the 5D manifold be interpreted as a conformally flat 4D space-time with superimposed electromagnetic fields. In this case, the foliated 4D α α α metric isg ˜µν = ηµν and we choose f µ to be the Kronecker-delta tensor; i.e., f µ = 1 µ. Furthermore, in 2 cgs units, we have Nµ = qAµ/c , where q is specific charge, Aµ is the electromagnetic potential, and c is the in vacuum.− We describe 5D null propagation by assigning two 5D null vectors (or their representations), one in- going and one outgoing, at each point of the 5D manifold. Point-to-point infinitesimal propagation may be described by infinitesimal translation followed by suitable, smooth isometric transformations of the null vectors. Hence, we impose smoothness conditions. We would like each be smoothly cast into any other null vector, by local isometry transformations.5 This is not possible since O(1,4) is not a connected . Hence, we represent each pair of 5D null vectors as a biquaternion,6 and describe 5D null propagation using a map Ψ : H1,1. The biquaternion Ψ(xM ) at point p, with coordinates xM , transforms with elements of Sp(1,1)M [i.e., → Ψ(xM ) SΨ(xM )], which is simply connected and compact. Next, we require the map Ψ(→xA) be Clifford differentiable [21] (i.e., differentiable with respect to x = A s A x ΓA) in every tangent space endowed with l 1,4 structure. This requirement constrains Ψ(x ) to satisfy the following differential equation C

Aa b A a Γ b∂AΨ (x )=0 . (3.13)

As it turns out, the constraint (3.13) is already sufficient for describing 5D particle propagation. We are A A particularly interested how Ψ(x ), solution of (3.13), behaves on the local null cone p = x p; x = C { ∈ T 4 † † A 2 The square norm of a Clifford element x is given by the product x x = x x = PA(x ) , inducing a 5D Euclidian metric. 5In the 4D picture, this would imply smooth transformations between particle and antiparticle states. 6A pair of 5D null vectors has the same of real degrees of freedom as a biquaternion.

3 A s 2 M x ΓA l 1,4 x = 0 . We may write the derivative in the local tangent space as ∂A = eA ∂M and cast Eq. (3.13)∈ C over| an open} neighborhood of the 5D manifold

Aa M b M a (Γ beA )∂M Ψ (x )=0 . (3.14)

This completes our spin description of 5D null propagation, as an alternative to the null path-integral formulation [15–17]. Dirac operators over curved manifolds are commonly introduced using standard Clifford algebras and spin connections [21]. These mathematical constructs do not appear in Eq. (3.14). In the applications we present below, we break wavefunction covariance and use metrics h˜MN that are independent of both x5 and x0. Hence, we derive 5D equations for quaternionic wavefunctions, starting from Eq. (3.14) rewritten as

5 α α Γ ∂5Ψ + Γ ∂αΨ + Γ Nα∂5Ψ =0, (3.15)

µ where Nα Nµf α. In the primed representation for the Γ matrices (Appendix A), we obtain ≡ j (∂5 + ∂0 + N0∂5)Ψ2 σ (∂j + Nj ∂5)Ψ1 =0, (3.16) − j (∂5 ∂0 N0∂5)Ψ1 + σ (∂j + Nj ∂5)Ψ2 =0, (3.17) − − j 7 where σ are the Pauli matrices and Ψ1,2 are the quaternionic components of Ψ. We impose ∂0Nα = 0 j and ∂5Nα = 0, and apply the operator σ (∂j + Nj ∂5) on the left of Eqs. (3.16) and (3.17), mindful of the

j j [σ (∂j + Nj ∂5), (∂5 ∂0 N0∂5)] = σ (∂j N0)∂5. (3.18) ± ± ± Introducing the wavefunctions Ψ = Ψ1 Ψ2, Eqs. (3.16) and (3.17) yield ± ± 2 2 j 2 j [∂ (∂0 + N0∂5) ]Ψ + [σ (∂j + Nj∂5)] Ψ σ (∂j N0)∂5Ψ =0. (3.19) 5 − ± ± ∓ ± It is interesting to contrast Eq. (3.19) to the corresponding 5D Klein-Gordon (KG) equation [15]

M 2 2 j j 5 ˜ ˜ Ψ=0 [∂ (∂0 + N0∂5) ]Ψ + [(∂ + N ∂ )(∂j + Nj ∂5)]Ψ = 0; (3.20) ▽M ▽ ⇔ 5 − ˜ ˜ M is the covariant derivative of the metric hMN (3.10) withg ˜µν = ηµν , and we imposed only ∂5Nµ = 0. ▽ 2 2 The operator ∂5 (∂0 + N0∂5) is key for the non-relativistic limit and common to both Eqs. (3.19) and (3.20), virtue to the− representation of the Γ matrices. The non-relativistic limit is thought as the limit of vanishing kinetic energy or low temperatures. For a particle propagating with null 5-momentum pA (i.e., a 5D photon) in flat 5D space-time, we impose pj p0 and pj p5 , which yield p0 p5. Hence, a particle starting at the origin of the coordinate frame| | ≪ remains | | localized| | ≪ | around| x0 x5.≈± The propagation problem reduces to the 5D KG equation, which, in light-cone coordinates, ≈±

y0 = x5 x0, yj = xj , y5 = (x0 + x5)/2, (3.21) − leads to the Schr¨odinger and the Fokker-Planck (FP) equations [15]. In the case of weak fields, Eq. (3.21) still provides approximate light-cone coordinates where

2 2 2 ∂ ∂ (∂0 + N0∂5) 2 2N0∂0∂5. (3.22) 5 − ≈ ∂y0∂y5 −

The operators ∂0 and ∂5 in Eqs. (3.19) and (3.22) may be explicitly written as ∂ 1 ∂ ∂ 1 ∂ ∂0 = + , ∂5 = + , (3.23) −∂y0 2 ∂y5 ∂y0 2 ∂y5 subject to further approximations, depending on the application.

7In this case, the Lorentz and Coulomb gauges are equivalent.

4 4 The Dirac equation

We now explain how Eq. (3.14) relates to the Dirac equation. The biquaternion Ψ can be cast into a bispinor pair; see Appendix B. Hence, solving Eq. (3.14) reduces to solving

M Aa b M a e AΓ b∂M ψ (x )=0 , (4.24) where ψ is a bispinor. Using Eq. (3.12), left-multiplying by i~Γ5, and performing a Fourier transform with respect to x5 (n.b., x5 is Fourier conjugated to λ−1 = mc/~), we obtain a Dirac-type equation describing propagation in a curved 4D space-time with superimposed electromagnetic fields

α µ γ (i~f α∂µ + qmAα/c)ψˆ mcψˆ = 0, (4.25) − where we used hatˆto denote the Fourier transforms of 5D wavefunctions with respect to x5. The matrices γα Γ5Γα form the basis for a foliated 4D Clifford algebra where ≡ αa βb βa αb αβ a γ bγ c + γ bγ c = 2η 1 c. (4.26) − In this case, rather than requiring Eq. (4.25) be defined on the 5D null structure, we impose an on-shell constraint for the propagation in the 4D space-time resulting from foliating x5. In general, this constraint 8 depends on the electromagnetic field. If the foliated 4D metric is flat (i.e.,g ˜µν = ηµν ) and we choose α α f µ =1 µ, then Eq. (4.25) becomes the Dirac equation [1] where the electromagnetic field enters according to the minimal coupling recipe

α γ (i~∂α + qmAα/c)ψˆ mcψˆ = 0. (4.27) − A gauge transformation of the electromagnetic field may be introduced in Eq. (3.14) through a 5D transformation of coordinates: xµ xµ, x5 x5 + (xµ). If the transformation is passive, it applies to the → → N 5D manifold coordinates, resulting in a transformation for the shift of the foliation Nµ Nµ + ∂µ , which is proportional to the electromagnetic field and satisfies the Maxwell equations [16]. If the→ transformationN is active, then one changes

Ψ(xµ, x5) Ψ(xµ, x5 + (xµ)). (4.28) → N A Fourier transform of Eq. (4.28) with respect to x5 yields

Ψˆ Ψˆ µ −1 Ψˆ qm µ exp[i (x )λ ] exp i ~ (x ) , (4.29) → N ≡ h− c A i where (xµ) is the gauge function of the electromagnetic field. Hence, as expected, the electromagnetic field can beA interpreted as a U(1)-gauge field for Eq. (4.25) and the Dirac equation (4.27), but not the 5D Dirac equation (3.14). This formalism may be developed further using the 4D Clifford algebra of the γ matrices (4.26) and the picture of a 4D space-time with superimposed electromagnetic field [23]. Derivations of the Dirac equation from first physics principles, in both flat and curved 4D space-times were previously discussed [22, 24–27]. Still, the Dirac equation has been criticized for the Klein paradox [28–31] and causality violation [32–35] in . The Klein paradox may be solved by imposing smooth interaction potentials (i.e., smooth space-time manifold) for the Dirac equation [29]. Our 5D approach revisits the geometric interpretation of the Dirac equation, bringing additional physics into play. First, the particle (passive) mass appears through Fourier transform with respect to x5. Second, electromagnetic (minimal) coupling enters naturally the Dirac equation, virtue to the 5D geometry. Furthermore, arguments of causality violation brought against the Dirac equation assume that propagation is fundamentally 4D, locally confined to the interior of 4D null cones. They do not hold when considering that the Dirac equation originates from a description of 5D null propagation.

8 α β 2 2 In each tangent space of the 4D manifold with electromagnetic fields, we have ηαβ p p = −m c . However, this result cannot be extended over the manifold without accounting for the electromagnetic field.

5 4.1 On the non-relativistic limit of the Dirac equation It is well-known that the non-relativistic limit of the Dirac equation results as the Pauli equation, when separating small and large components of the bispinor wavefunction ψˆ [36,37]. Here, we do not employ this technique. Rather, we use 5D geometric-based approximations to obtain equations for the non-relativistic propagation of a quantum particle (i.e., p5 > 0) in time-independent, weak fields. We consider that y5 x5 ≈ and p0 p5 (i.e., ∂0Ψ ∂5Ψ ) in the terms containing fields in Eqs. (3.19) and (3.22) to obtain ≈ ± ≈ ±

∂ 2 j 2 j 2 ∂5 N0∂ Ψ + [σ (∂j + Nj∂5)] Ψ σ (∂j N0)∂5Ψ 0. (4.30) ∂y0 − 5  ± ± ∓ ± ≈

We perform a Fourier transform of Eq. (4.30) with respect to x5 and use y0 cτ as time coordinate, conjugated with the mechanical energy, only; i.e., not including mc2. Furthermore,≡ we cast the wavefunction Ψˆ ± into a pair (Appendix B) for a non-relativistic picture of spin 1/2 quantum mechanics, based 3 3 on the L2(R ) L2(R ) Hilbert space equipped with the canonical dot product. In typical notation of 3D physics, this yields×

~ ˆ ~ 2 ∂ψ± 1 e A~ ψˆ + eA0ψˆ µσ~ B~ ψˆ iµσ~ E~ ψˆ , (4.31) − i ∂τ ≈ 2m  i ∇− c  ± ± − ± ∓ ±

ˆ ~ ~ ~ where ψ± is the spinor wavefunction, B = A and E = A0 are time-independent magnetic and electric fields, and e qm and µ q~/(2c) are,∇× respectively, the−∇ particle’s charge and magneton. Equation ≡ ≡ ~ ˆ (4.31) represents the Pauli equation for time-independent settings, except for the term iµσ~ Eψ±, describing an interaction between spin and time-independent electric field. An anti-hermitian spin-electric∓ interaction term occurs next to the well-known Stern-Gerlach (hermitian) spin-magnetic term. Thus, the total spin- ~ ~ ˆ ~ ~ interaction term is given by µσ~ (B iE)ψ±, where B iE is reminiscent of Riemann-Silberstein fields [38,39] and the photon wavefunction problem± [40, 41]. ± Spin-electric interactions are common in the phenomenology of condensed matter physics. They received much attention [42, 43], particularly with the discovery of the electric-dipole spin resonance [44, 45]. Spin- orbit coupling and hyperfine interactions can mediate spin-electric interactions. There are even cases where the Stern-Gerlach term contributes toward spin-electric interactions [46]. In contrast to phenomenological ~ ˆ contributions, iµσ~ Eψ± has fundamental status and is anti-hermitian, conventionally describing decoher- ence, decay and∓ resonant scattering induced by electric fields on spin 1/2 particles. In fact, the process of breaking covariance of the Dirac equation (4.27) down to Eq. (4.31) can be naturally explained in the language of non-hermitian quantum mechanics [47–49]. The Dirac operator is essential self-adjoint [50–54], leading to real energy spectra for many field config- urations. Hence, the Dirac particle, described by the bispinor wavefunction ψˆ , represents a closed system. ˆ ˆ 9 Due to covariance breaking, ψ is rearranged into the spinor wavefunctions ψ±. This results into two open systems, defined by non-hermitian hamiltonians on smaller Hilbert spaces. Each of these systems undergoes decoherence and consists of a Pauli particle in interaction with the environment through an anti-hermitian spin-electric term. However, the two open systems complete each other to form a larger closed system, which remains coherent. From the perspective of the Pauli equation, Eq. (4.31) deserves attention in its own right. We discuss a simple application, neglecting the effect of electromagnetic fields on particle orbits, and analyzing only the spin degrees of freedom. Equation (4.31) yields

~ ∂ψˆ µσ~ (B~ + iE~ )ψˆ Hψˆ , (4.32) − i ∂τ ≈− ≡

where H is the hamiltonian operator acting on ψˆ . We analyze Eq. (4.32) using the quantum theory of decay processes [55], but other approaches [56] may apply, as well. The probability10 that state ψˆ , prepared

9 ˆ For exact covariance breaking, without taking the non-relativistic limit, consider spinor solutions ψ± of the Fourier trans- form with respect to x5 of Eq. (3.19). 10This quantity is not strictly a probability when hamiltonians are non-hermitian, since its value may exceed 1.

6 at τ = 0, remains unaltered is given by

2 ˆ † ~ ˆ Pψˆ(τ)= ψ exp( iHτ/ ) ψ . (4.33) −

Thus, within an ensemble of identical particles, the number N (τ) of particles in state ψˆ at time τ is ˆ N(τ) = N(0)Pψˆ(τ). If Pψˆ(τ) decreases with τ, then we say that ψ undergoes spontaneous decay. If Pψˆ(τ) increases with τ, then we say that ψˆ gets populated. In our case, we have iσ~ ~ω exp( iHτ/~) = cos(ωτ)+ sin(ωτ), (4.34) − ω where ~ω µ(B~ + iE~ )/~ and ω (~ω)2. When E~ adds just a small perturbation to spin precession in the ≡ ≡ magnetic field B~ , the first order expansionp in E~ / B~ yields ω µ( B~ + i E~ cos θ)/~, where θ is the angle | | | | ≈ | | | | between E~ and B~ . The probabilities that spin up (+) and spin down ( ) states, prepared with the magnetic − field B~ , remain unaltered are

P (τ) = exp( 2 Im(ω)τ) = exp( 2µ E~ cos θτ/~). (4.35) ± ± ± | | Hence, spontaneous quantum transitions, due to the perturbing electric field E~ , occur on the time scale

2 Im(ω) −1 = ~/(2µ E~ cos θ ), (4.36) | | | || | which determines the coherence time and spectral width of the transition between spinor states prepared with the field B~ . These transitions can be observed in resonant regime by augmenting the experimental setup corresponding to Eq. (4.32) with the presence of a electromagnetic radiation field, for photons to be absorbed and reemited at the Larmor frequency, µ B~ /~. This constitutes, in fact, the setup of electric-field nuclear magnetic resonance [57,58]. However,| it| appears that resonance broadening versus external electric field has not been addressed systematically. Nevertheless, particles in condensed state are often subject to internal, local electric fields. It is well established that electric quadrupole interactions strongly broaden the nuclear magnetic resonance [59], in qualitative aggreement with Eq. (4.36).

5 On the 5D Dirac equation in statistical physics

Quantum propagation has been previously invoked for a space-time picture of statistical physics. The original formalism, proposed by Feynman [60], is based on path-integrals in a 4D space-time with Riemannian metric, where Kelvin temperature is used to construct the time coordinate; see Ref. [61] for a recent review. In this case, the path integral between two arbitrary space-time events represents the propagator of a diffusion equation, which is interpreted as a quantum equation. Hence, the path integral is not indispensible and the formalism can be fully recovered from the diffusion equation alone. Here, general considerations and Eq. (3.14) yield a single-particle formalism for spin 1/2, in line with ear- lier work on 5D spinless propagation [15,16]. We start with the observation that thermodynamic observables at thermal equilibrium do not change with time. We assume that a particle achieves thermal equilibrium for a time-independent metric and impose h˜MN (3.10) be independent of both x5 and x0. In this case, the 5D metric can be foliated along x5 or x0, so physics can be interpreted from the perspective of mechanics or statistics, respectively. The coordinate transformations that preserve this property are dipheomorphisms of 3D space (n.b., the local isometry group is O(3)). We further assume that a particle at thermal equilibrium is always found at rest by quantum measurement. For connecting with previous results and mathematical convenience, we analyze the non-relativistic dy- namics of a statistical particle in the presence of weak fields. Combining Eqs. (3.19) and (3.22), we obtain

2 ∂ j 2 j 2 N0∂0∂5 Ψ + [σ (∂j + Nj ∂5)] Ψ σ (∂j N0)∂5Ψ 0. (5.37) ∂y0∂y5 −  ± ± ∓ ± ≈

7 We introduce spinor rather than quaternionic wavefunctions, perform a Laplace transform with respect 0 5 0 j to y , and give y cu the meaning of time. Thus, searching for solutions in the form ψ±(y , x ,u) = 0 ′ j ≡ ′ j exp( y /Λ)ψ±(x ,u) yields equations for ψ±(x ,u), possibly with non-hermitian hamiltonians, where evo- lution− is described by the dilatation group D(1), rather than U(1). Hence, these equations are interpreted as diffusion equations, where temperature enters indirectly through an empirical model for the diffusion con- stant. The fundamental solutions of the diffusion equations yield non-relativistic, single-particle propagators ′ from (~x, 0) to (~x, u), denoted ψ±(~x, 0; ~x, u). We use traces of these propagators over the space and spin degrees of freedom to define the Massieu function, Ψ/kB, for a single particle at constant temperature

′ ′ Ψ/kB = ln d~x [Tr ψ+(~x, 0; ~x, u)+Tr ψ (~x, 0; ~x, u)], (5.38) Z − ′ ′ where the sum between Tr ψ+(~x, 0; ~x, u) and Tr ψ−(~x, 0; ~x, u) represents the trace over the spinor degrees of freedom. Thermodynamics require Ψ/kB be real-valued, which is not granted by Eq. (5.38) and remains to be established for each particular application. When the Massieu function is well defined, thermodynamic results may be used to identify u in Eq. (5.38) as a time scale specific to the statistical particle.11 This is very different from current developments in non-hermitian quantum thermodynamics [62, 63], where the class of hamiltonians is carefully restricted such that the proposed formalism does not encounter difficulties. Many-particle statistics may be approached through an ergodic principle. We argue heuristically that single-particle propagation for the time interval Nu is statistically equivalent, in terms of suitably defined observables, to that of an ensemble of N distinguishable, non-interacting particles, for a time interval u [16]. A major limitation of the formalism described in this section is that it addresses only the thermodynamics of particle propagation. This may not be sufficient to account for all temperature dependence observed in experiment, because the space-time metric may depend on temperature, as well.12 This temperature dependence may be included empirically in the metric; here, it remains unaccounted for. While the role of temperature for the propagation problem defined by Eqs. (5.37) and (5.38) may not be fully resolved, our approach allows for addressing other physical aspects, such as the thermodynamic impact of external fields. We discuss two physical systems to clarify the use of our formalism.

5.1 The free particle For the flat 5D metric, the spin degrees of freedom decouple and may be ignored; Eq. (5.37) reduces to an equation for a scalar function ψ. A Laplace transform with respect to y0 yields ∂ cΛ ψ′ = 2ψ′. (5.39) ∂u 2 ∇ Taking Eq. (5.39) for the FP equation relates Λ to the diffusion coefficient; i.e., D = cΛ/2. To link the formalism to thermodynamics, we introduce Kelvin temperature indirectly, through an empirical model of the diffusion coefficient as a function of temperature, T . Here, we invoke the Stokes-Einstein model D =1/(βζ), where β =1/(kBT ) and ζ is the drag coefficient. The FP equation typically stands for the continuity equation of the probability density to localize a Brownian particle of mass m, undergoing non-relativistic dynamics in an environment with temperature T and drag ζ, according to the Langevin equation. We reinterpret the FP equation as the wave equation for a statistical particle of mass M = ~/(2D) (i.e., Λ−1 Mc/~), with the advantage of more flexibility in adding fields to Eq. (5.39). ≡ ′ j ~ ′′ j 2 ′′ j ~ The wavefunctions ψ (x ,u) = exp( e~v u/ )ψ~v (x ), where e~v = M~v /2 and ψ~v (x ) = exp(iM~v~x/ ), are eigenstates of 2 in Eq. (5.39). Thus,− we may write [16] ∇ ′ ′′ j ′′ j † ψ (~x, 0; ~x, u)= d~vψ (x ) exp( e~v u/~)[ψ (x )] = d~v exp( e~v u/~), (5.40) Z ~v − ~v Z − 11The time-scale u may be thought as the minimum amount of time for which a particle should be observed in order to be assigned thermodynamical observables. In Ref. [16], this is called a quantum of physical time. 12For example, consider the model by which electrons propagate within metal at temperature T . It is natural to assume that the background of electron propagation depends on temperature, which also determines the thermal of the electrons.

8 and

′ Ψ/kB = ln d~xψ (~x, 0; ~x, u). (5.41) Z

2 We choose u =2m(βD)=2m/ζ [16], such that e~v u/~ = βm~v /2, and Ψ/kB yields the traditional thermo- dynamics for an ideal gas consisting of a single particle.

5.2 Low-temperature spin in uniform electric and magnetic fields Similarly to Eq. (4.32), we consider weak fields that affect only particle’s spin. We invoke the Arrhenius diffusion model D exp( TD/T ) to explicitly request particle diffusion be negligible; i.e., T TD. Still, thermal energy may∝ be stored− by the spin degrees of freedom. Under these conditions, Eq. (5.37)≪ reduces to

∂2ψ q ∂ 1 ∂ 2 ± σ~ (iB~ E~ ) + ψ 0. (5.42) ∂y0∂y5 − c2 ∓ ∂y0 2 ∂y5  ± ≈

A Laplace transform with respect to y0 yields

ψ′ σ ~ ~ ψ′ ~∂ ± µ~ (iB E) ± ~ ~ ′ ′ ∓ µσ~ (iB E)ψ± hψ±. (5.43) − ∂u ≈−1+(Λ/2)q/(2c2)σ~ (iB~ E~ ) ≈− ∓ ≡ ∓ Hence, we formaly recover Eq. (4.32), where we now require D(1) evolution. The hamiltonian h = µσ~ (iB~ − ∓ E~ ) is non-hermitian. The hermitian part of h yields a coherent quantum picture, while the anti-hermitian part of h contributes as a perturbation leading to decoherence. We say that, within the statistical picture, the electric field triggers a quantum coherent response from the particle’s spin, while the magnetic field leads to decoherence of the spin state.13 The converse is valid in the quantum mechanical picture. The propagator of Eq. (5.43) may be written as

σ~ Ω~ ± exp( hu/~) = exp( σ~ Ω~ ±u) = cosh(Ω±u) sinh(Ω±u), (5.44) − − − Ω±

2 where Ω~ ± µ( iB~ E~ )/~ and Ω± (Ω~ ±) µ( E~ i B~ cos θ)/~. Taking the trace of the propagator over the spin≡ and− spinor± degrees of freedom,≡ q we≈ obtain| |∓ | |

µu ~ µu ~ Ψ/kB = ln [cosh(Ω+u) + cosh(Ω−u)] = ln 2 cosh ~ E cos ~ B cos θ . (5.45) h  | |  | | i The electric and magnetic susceptibilities of the spin 1/2 particle are given by

2 2 1 ∂ (Ψ/kB) 1 µu χe = = , (5.46) 2 ~ 4πβ ∂ E~ 4πβ   | | |E~ |=0

2 2 1 ∂ (Ψ/kB) 1 µu 2 χm = = cos θ, (5.47) 2 ~ 4πβ ∂ B~ −4πβ   | | |B~ |=0

respectively; the case E~ = 0 is formally recovered in Eq. (5.47) by substituting cos2 θ with 1. Hence, the non-diffusing spin| 1/2| particle is paraelectric and diamagnetic, where the time-scale u remains to be established phenomenologically. One possibility is to use Feynman’s recipe, u = β~. However, other choices may be better motivated in applications, and offer a more adapted thermodynamical formalism, particularly that we required T TD. We also required the electric and magnetic fields be sufficiently weak, so their effect on particle orbital≪ motion is negligible. If any of these requirements are violated, the particle may lose its thermodynamic properties, as expressed by Eqs. (5.46) and (5.47).

13The magnetic field leads to non-hermitian hamiltonians in the statistical pictures of the KG [15] and Schr¨odinger [16] equations, as well.

9 The above results may be important for current discussions on superconductivity. The reference theory, proposed by Bardeen, Cooper and Schriffer [64–66], has been challenged for its interpretation of the Meissner effect [67, 68] and the conduction mechanism [69]. The single-particle thermodynamics developed in this section provide a new starting point to explain the Meissner effect, based on statistical physics with non- hermitian hamiltonians. Furthermore, the superconduction mechanism may rely on interactions between spins of low-temperature particles that are aligned by an external electric field (i.e., electric spin waves). These remain topics for further work.

6 Discussion and conclusions

We demonstrated how a 5D formalism of null propagation can provide space-time pictures of spin-1/2, 4D quantum and statistical mechanics. The resulting 4D hamiltonians in quantum and statistical mechanics may be distinct and non-hermitian. Still, coherent quantum pictures are provided by the hermitian parts of these hamiltonians, where the anti-hermitian parts may be considered as perturbations. Our statistical formalism appears to go against Pauli’s theory of paramagnetism. However, we note that the phenomenology associated with our statistical formalism may be significantly richer than what we presented here. The minimal requirement for a space-time picture of statistical physics is that the 5D metric be indepen- dent of x0 [16]. Strictly speaking, it is not necessary that the 5D metric be independent of x5 and, implicitly, the mass m of the quantum particle be a well-defined observable. It is sufficent that the mass M, or the concept of diffusion coefficient, be well defined. Under these circumstances, it is natural to foliate the 5D manifold along x0 and consider the role played by the shift of this foliation in statistical mechanics. These fields, included in h0M , are key for the Lense-Thirring precession and subject of the gravito-electromagnetism theory. Their microscopic role for statistical phenomena may be carefully considered. A well-rounded statistical theory may benefit the understanding of other physical phenomena besides superconductivity. Particularly, it may benefit the study of thermodynamic response of condensed matter to external electromagnetic fields. On the quantum mechanical side, the phenomenology of the Dirac equation is well understood. Still, experimental confirmation of the anti-hermitian term of spin-electric interaction may provide insight into how to control the coherence time of quantum devices.

References

[1] P A M Dirac. The quantum theory of the electron. Proceedings of the Royal Society of London. Series A, 117(778):610–624, 1928. [2] P A M Dirac. The quantum theory of the electron. Part II. Proceedings of the Royal Society of London. Series A, 118(779):351–361, 1928. [3] J A Schouten and D van Dantzig. Generelle Feldtheorie. Zeitschrift fur Physik, 78(9-10):639–667, 1932. [4] W Pauli. Uber¨ die Formulierung der Naturgesetze mit f¨unf homogenen Koordinaten. Teil II: Die Dirac- schen Gleichungen f¨ur die Materiewellen. Annalen der Physik, 410(4):337–372, 1933. [5] P A M Dirac. The electron wave equation in de-Sitter space. Annals of Mathematics, 36(3):657–669, 1935. [6] J Lubanski. Sur la th´eorie des particules ´el´ementaires de spin quelconque. I. Physica, 9(3):310–324, 1942. [7] J Lubanski. Sur la th´eorie des particules ´el´ementaires de spin quelconque. II. Physica, 9(3):325–338, 1942. [8] H J Bhabha. Relativistic wave equations for the elementary particles. Reviews of Modern Physics, 17(2-3):200–216, 1945. [9] F G¨ursey and T Lee. Spin 1/2 wave equation in De-Sitter Space. Proceedings of the National Academy of Sciences of the United States of America, 49:179–186, 1963.

10 [10] J Kocinski. A five-dimensional form of the Dirac equation. Journal of Physics A, 32:4257–4277, 1999. [11] G M Zhang, Y L Wu, X Z Wang, and J L Sun. Dirac field of Kaluza-Klein theory in Weitzenbock space. International Journal of Theoretical Physics, 39(8):2051–2062, 2000. [12] N Redington and M A K Lodhi. Simple five-dimensional wave equation for a Dirac particle. Journal of , 48:022303, 2007. [13] A O Barut. Dynamical symmetry group based on Dirac equation and its generalization to elementary particles. Physical Review, 135(3B):B839–B842, 1964. [14] A J Bracken and H A Cohen. On canonical SO(4, 1) transformations of the Dirac equation. Journal of Mathematical Physics, 10(11):2024–2032, 1969. [15] R Breban. A five-dimensional perspective on the Klein–Gordon equation. Annals of Physics, 356:158– 170, 2015. [16] R Breban. Interpretation of the five dimensional quantum propagation of a spinless massless particle. Progress of Theoretical Physics, 114(3):643–668, 2005. [17] R Breban. On spinless null propagation in five-dimensional space-times with approximate space-like Killing symmetry. The European Physical Journal C, 76(9):486–4, 2016. [18] M Jammer. Concepts of mass in classical and modern physics. Courier Corporation, 1997. [19] M Jammer. Concepts of mass in contemporary physics and philosophy. Princeton University Press, 2009. [20] P S Wesson. Space-Time-Matter. World Scientific, 1999. [21] J E Gilbert and M Murray. Clifford algebras and Dirac operators in harmonic analysis. Cambridge University Press, 1991. [22] R M Wald. . University of Chicago Press, 2010. [23] B Thaller. The Dirac Equation. Springer Science & Business Media, Berlin, Heidelberg, 2013. [24] S K Srinivasan and E C G Sudarshan. A direct derivation of the Dirac equation via quaternion measures. Journal of Physics A, 29(16):5181–5186, 1996. [25] H P Noyes. Finite and discrete model for single fermion mass renormalization: Derivation of the free particle Dirac equation. Physics Essays, 8(3):434, 1995. [26] Y Jack Ng and H van Dam. A geometrical derivation of the Dirac equation. Physics Letters A, 309(5- 6):335–339, 2003. [27] A Deriglazov and A Nersessian. Rigid particle revisited: Extrinsic curvature yields the Dirac equation. Physics Letters A, 378(18-19):1224–1227, 2014. [28] A D Alhaidari. Resolution of the Klein paradox. Physica Scripta, 83(2):025001, 2011. [29] A Calogeracos and N Dombey. History and physics of the Klein paradox. Contemporary Physics, 40(5):313–321, 1999. [30] K Miyazaki. Validity and limitation of the Dirac phenomenology. Progress of Theoretical Physics, 89(3):639–655, 1993. [31] E Truebenbacher. The Klein paradox: A new treatment. European Journal of Physics, 36(5):055015, 2015. [32] N Barat and J C Kimball. Localization and causality for a free particle. Physics Letters A, 308(2- 3):110–115, 2003.

11 [33] D P L Castrigiano and A D Leiseifer. Causal localizations in relativistic quantum mechanics. Journal of Mathematical Physics, 56(7):072301–35, 2015. [34] G Hegerfeldt. Violation of causality in relativistic quantum theory? Physical Review Letters, 54(22):2395–2398, 1985. [35] B Rosenstein and M Usher. Explicit illustration of causality violation: Noncausal relativistic wave- packet evolution. Physical Review D, 36(8):2381–2384, 1987. [36] F Schwabl. Advanced Quantum Mechanics. Springer-Verlag 2005. [37] K T Hecht. Quantum Mechanics. Springer-Verlag 2000. [38] L Silberstein. Elektromagnetische Grundgleichungen in bivectorieller Behandlung. Annalen der Physik, 327:579–586, 1907. [39] I Bialynicki-Birula and Z. Bialynicka-Birula. The role of the Riemann-Silberstein vector in classical and quantum theories of electromagnetism. Journal of Physics A, 46(5):053001, 2013. [40] I Bialynicki-Birula. On the of the photon. Acta Physica Polonica A, 86(1-2):97-116, 1994. [41] I-I Popescu, P Sterian and M Dobre. The photon wave function and the Fresnel formulas. Romanian Reports in Physics, 62(2):360–368, 2010. [42] E Rashba. Theory of electric dipole spin resonance in quantum dots: Mean field theory with Gaussian fluctuations and beyond. Physical Review B, 78(19):195302, 2008. [43] R Li, J Q You, C P Sun and F Nori. Controlling a nanowire spin-orbit qubit via electric-dipole spin resonance. Physical Review Letters, 111(8):086805, 2013. [44] S Nadj-Perge, S M Frolov, E P A M Bakkers and L P Kouwenhoven. Spin-orbit qubit in a semiconductor nanowire. Nature, 468(7327): 1084–1087, 2010. [45] H Yuan, M S Bahramy, K Morimoto, S Wu, K Nomura, B J Yang, H Shimotani, R Suzuki, M Toh, C Kloc, X Xu, R Arita, N Nagaosa, and Y Iwasa. Zeeman-type spin splitting controlled by an electric field. Nature Physics, 9(9):563–569, 2013. [46] J Ramrez-Ruiz, S Boutin, I Garate. NMR in an electric field: A bulk probe of the hidden spin and orbital polarizations. Physical Review B, 96(23):235201, 2017. [47] A Rivas and S F Huelga. Open quantum systems: An introduction. Springer, Berlin, 2012. [48] N Moiseyev. Non-hermitian quantum mechanics. Cambridge University Press, 2011. [49] M Castagnino and S Fortin. Non-hermitian hamiltonians in decoherence and equilibrium theory. Journal of Physics A, 45(44):444009, 2012. [50] M Arai. On essential selfadjointness, distinguished selfadjoint extension and essential spectrum of Dirac operators with matrix valued potentials. Publications of the Research Institute for Mathematical Sciences, 19(1):33–57, 1983. [51] N Arrizabalaga, J Duoandikoetxea, and L Vega. Self-adjoint extensions of Dirac operators with Coulomb type singularity. Journal of Mathematical Physics, 54(4):041504, 2013. [52] V Georgescu and A Iftimovici. Localizations at infinity and essential spectrum of quantum hamiltonians I: General theory. Reviews in Mathematical Physics, 18(4):417–483, 2006. [53] B G¨uneysu and O Post. Path integrals and the essential self-adjointness of differential operators on noncompact manifolds. Mathematische Zeitschrift, 275(1-2):331–348, 2013.

12 [54] H Kalf and O Yamada. Essential self-adjointness of Dirac operators with a variable mass term. Pro- ceedings of the Japan Academy, Series A, 76(2):13–15, 2000. [55] L Fonda, G C Ghirardi, and A Rimini. Decay theory of unstable quantum systems. Reports on Progress in Physics, 41(4):587–631, 1978. [56] K G Zloshchastiev. Non-hermitian hamiltonians and stability of pure states. European Physical Journal D, 69(11):024110, 2015. [57] B H Ruessink and C MacLean. Electric field nuclear magnetic resonance (application to nuclear quadrupole coupling). Zeitschrift f¨ur Naturforschung A, 41(1-2):421–424, 1986. [58] C W Hilbers and C MacLean. Electric field effects in nuclear magnetic resonance. Molecular Physics, 16(3):275–284, 1969. [59] D Freude. Quadrupolar nuclei in solid-state nuclear magnetic resonance. Encyclopedia of Analytical Chemistry (Eds. R A Meyers and C Dybowski). John Wiley & Sons, Chichester, 2006. [60] R P Feynman, A R Hibbs, and D F Styer. Quantum mechanics and path integrals. Courier Corporation, New York, 2010. [61] K-Y Wong. Review of Feynman’s path integral in quantum statistics: From the molecular Schr¨odinger equation to Kleinert’s variational perturbation theory. Communications in Computational Physics, 15(4):853–894, 2014. [62] H F Jones, and E S Jr Moreira. Quantum and classical statistical mechanics of a class of non-hermitian hamiltonians. Journal of Physics A, 43(5):055307, 2010. [63] B Gardas, S Deffner and A Saxena Non-hermitian quantum thermodynamics. Scientific Reports, 6(1):23408, 2016.

[64] J Bardeen, L N Cooper and J R Schrieffer. Microscopic theory of superconductivity. Physical Review, 106(1):162–164, 1957. [65] J Bardeen, L N Cooper and J R Schrieffer. Theory of superconductivity. Physical Review, 108(5):1175– 1204, 1957. [66] J R Schrieffer. Theory of superconductivity. CRC Press, 1964. [67] J E Hirsch. Correcting 100 years of misunderstanding: electric fields in superconductors, hole supercon- ductivity, and the Meissner effect. Journal of Superconductivity and Novel Magnetism, 25(5):1357–1360, 2012. [68] J E Hirsch. Momentum of superconducting electrons and the explanation of the Meissner effect. Physical Review B, 95(1):333, 2017. [69] P Monthoux, A V Balatsky and D Pines. Toward a theory of high-temperature superconductivity in the antiferromagnetically correlated cuprate oxides. Physical Review Letters, 67(24):3448–3451, 1991. [70] S Helgason. Differential geometry, Lie groups, and symmetric spaces. Graduate Studies in Mathematics. Academic Press, New York, 1979.

13 A Two representations for the Γ matrices

Sp(1,1) is a pseudo-simplectic group whose generators can be represented in the following form14

x b a c  b∗ x c∗ a∗  M = , (A.48) a∗ −c− y d  −   c∗ a d∗ y   − − −  where x, y R, a,b,c,d C, and is symbol for complex conjugation. The Sp(1,1) generators with a = c =0 are generators∈ of SU(2) ∈ SU(2),∗ of Sp(1,1), which is a double cover of O(4). It is straightforward × to write four Γ-matrices that generate the corresponding l4 Clifford subalgebra C j 0 iσ 5 0 1 Γj = , Γ = − , (A.49)  iσj 0   1 0  − − where σj are the Pauli matrices (j, k, ... = 1, 2, 3) and iσj is a representation of the quaternion units. The Γ matrices may be considered 2 2 with complex quaternion entries or 4 4 with complex entries. They αa βb βa αb ×αβ a × satisfy Γ bΓ c +Γ bΓ c = 2(1 1 c), where the indices α, β = 1, 2, 3, 5 and a,b,c = 1, 2 or 1, 2, 3, 4, as needed. The generators of SU(2) SU(2) can be represented using of Γ matrices × l jk j k jk σ 0 σ i[Γ , Γ ]/2 = ǫ l , (A.50) ≡ −  0 σl  σj 0 σj5 i[Γj , Γ5]/2 = . (A.51) ≡  0 σj  − We now introduce 1 0 Γ0 iΓ1Γ2Γ3Γ5 = i . (A.52) ≡  0 1  − Aa Bb Ba Ab AB a Direct calculations provide Γ bΓ c +Γ bΓ c =2η 1 c, and the following commutation relations

j 0 0 0 iσ σ j i[Γ , Γj ]/2 = − , (A.53) ≡  iσj 0  − 0 1 σ05 i[Γ0, Γ5]/2 = , (A.54) ≡  1 0  − which are the other four Sp(1,1) generators, obtained by taking b = d = x = y = 0 in (A.48). The Γ matrices given by Eqs. (A.49) and (A.52) satisfy

Tr (ΓA)=0, (A.55) and have the desired properties under hermitian conjugation

† † † Γ = Γ0, Γ = Γj, Γ = Γ5. (A.56) 0 − j 5 s 0 Thus, Eqs. (A.49) and (A.52) provide a representation of the basis of l 1,4. Finally, defining Γ ipΓ + 1 2 3 5 † C ≡ qΓ Γ Γ Γ = ipΓ0 + qΓ1Γ2Γ3Γ5, where p, q R (n.b., Γ = Γ), we obtain − ∈ † ΓΓ = ΓAΓ (A.57) A − and

(Γ)2 = (p q)21. (A.58) − 14We permuted columns 2 and 3, and lines 2 and 3 in the matrix representation of Sp(1,1) generators provided by Ref. [70].

14 For applications, we note that the matrices γα Γ5Γα, given by ≡ 0 i1 iσj 0 γ0 = , γj = , (A.59)  i1 0   0 iσj  − − αa βb βa αb αβ a form the basis for a 4D Clifford algebra where γ bγ c + γ bγ c = 2η 1 c and α, β = 0, 1, 2, 3. Furthermore, for covariance breaking, we may use a different representation− of the Γ matrices, generated by conjugacy Γ′A = UΓAU −1 with the unitary matrix

1 1 i1 U = − . (A.60) √2  i1 1  − We obtain

j 0 0 1 σ 0 5 0 1 Γ′ = , Γ′j = − , Γ′ = . (A.61)  1 0   0 σj   1 0  − The matrices γ′α Γ′5Γ′α are given by ≡ j 0 1 0 0 σ γ′ = − , γ′j = . (A.62)  0 1   σj 0  − In the main text, we refer to this representation as the primed representation of the gamma matrices. The Dirac γ-matrices are obtained from Eq. (A.62) by changing the sign of γ′0; i.e., γ′0 γ′0. →− B The biquaternion-bispinor relationship

Given a bispinor ψ1, there exists a unique biquaternion Ψ which, represented with the hypercomplex units j ∗ iσ , has the bispinor ψ1 as its first column; i.e., Ψ = (ψ1 ψ2). Straightforward calculations yield ψ2 = Kψ1, where is symbol for complex conjugation, and K is a 4 4 antisymmetric matrix with real entries ∗ × 0 10 0 − 2  1 0 0 0  σ 0 K = = i . (B.63) 0 0 0 1 −  0 σ2   −   0 0 1 0    A biquaternion of unit norm

† 1 0 Ψ Ψ ψ1 Γ ψ1 ψ1 ψ1 ψ2 = † (ψ1 ψ2)= · · = , (B.64) ·  ψ   ψ2 ψ1 ψ2 ψ2   0 1  2 · · is equivalent to two mutually orthonormal bispinors ψ and ψ ; n.b., ψ ψ = 0 is identically satisfied, 1 2 1 · 2 owing to the relationship between ψ1 and ψ2. A quaternion-spinor relationship can be established in similar fashion. For each spinor ψ11, there exists j a unique quaternion Ψ1 which, represented with the hypercomplex units iσ , has the spinor ψ11 as its first 2 ∗ † column; i.e., Ψ1 = (ψ ψ ), where ψ = iσ ψ so ψ ψ =0. If Ψ1 has unit norm, we have 11 12 12 − 11 11 12 † Ψ Ψ Ψ†Ψ ψ11 1 0 1 1 = 1 1 = † (ψ11 ψ12)= , (B.65) ·  ψ12   0 1  implying that ψ11 and ψ12 are mutually orthogonal and have unit norm, as well.

15