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The Four Dimensional Dirac Equation in Five Dimensions

The Dirac equation may be thought as originating from a theory of five-dimensional (5D) space-time. We define a special 5D Clifford algebra and introduce a spin-1/2 constraint equation to describe null propagation in a 5D space-time manifold. We explain how the 5D null formalism breaks down to four dimensions to recover two single-particle theories. Namely, we obtain Dirac's relativistic quantum mechanics and a formulation of statistical mechanics. Exploring the non-relativistic limit in five and four dimensions, we identify a new spin-electric interaction with possible applications to magnetic resonance spectroscopy (within quantum mechanics) and superconductivity (within statistical mechanics).

In this work, we define a special 5D Clifford algebra using the O(1,4) symmetry and propose a constraint equation for 5D null propagation. We then use a recent interpretation of the 5D geometry [15][16][17] where a 5D null constraint yields a 4D on-shell constraint. In other words, the norm of the 5-momentum of a 5D photon in flat space-time η AB p A p B = 0 [A, B = 0, 1, 2, 3, 5, and η AB = diag(−1, 1, 1, 1, 1)] yields the 4D on-shell constraint for a particle of mass m, η αβ p α p β = −m 2 c 2 (α, β = 0, 1, 2, 3), where we used p 5 ≡ mc. The mass m is thought here as being passive [18,19], a dynamical variable which is not a source of gravitational field. 1 With this interpretation, the inverse Compton wavelength λ −1 = mc/ and the fifth coordinate x 5 are conjugated by Fourier transform. Furthermore, it is postulated that, although the geometry is 5D, the perception of an observer remains 4D. However, the 4D observer has two major tools for understanding the 5D space-time. If the 5D geometry is independent of the fifth coordinate, then the 5D physics can be interpreted as 4D quantum mechanics. If the 5D geometry is independent of time (i.e., the zeroth coordinate), then the 5D physics can be interpreted as 4D statistical mechanics [15][16][17]. Both mechanics and statistics pictures apply when the geometry is independent on both time and fifth coordinate. In what follows, we consider how a 5D equation delivers the Dirac equation in 4D quantum mechanics and applications to 4D statistical mechanics.
We are particularly interested in a certain class of curved 5D space-times with the 4D interpretation of a Minkovski space with superimposed electromagnetic field. These space-times should be regarded as local approximations of more realistic 5D geometries, including non-trivial gravitational fields, whose metrics satisfy suitable field equations [16,20]. The Dirac equation corresponding to these curved 5D space-times is well known [1] and tested in experiment. Hence, our discussion of the Dirac equation within Kaluza-Klein theory puts into perspective generalizations of the Dirac equation for curved manifolds [21,22].
The structure of this paper is as follows. In Sec. 2 we introduce a special Clifford algebra for a 5D flat space-time. In Sec. 3 we discuss a class of 5D curved space-times and postulate the description of 5D null propagation. In Secs. 4 and 5, we discuss the relation with the 4D formalism and work out examples of physical systems. In Sec. 6 we conclude our work.

A special 5D Clifford algebra
We use bold face to indicate that some indices of the corresponding mathematical objects are suppressed. Consider the vector space consisting of quaternion vectors with two components, henceforth called biquaternions. The scalar product of two biquaternions Ψ, Φ ∈ H 1,1 is defined by The symbol † stands for hermitian conjugation and Γ ab (a, b, ... = 1, 2 or 1, 2, 3, 4, as needed) 2 for a hermitian matrix with quaternion entries and signature diag(1, −1). The isometry group of H 1,1 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)/Z 2 ), we introduce a special Clifford algebra, Cl s 1,4 , such that its basis elements Γ Aa b satisfy the following relations where i is the complex unit, σ AB 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 We note that the Γ matrices are not determined by Eqs. (2.3)-(2.6), since a conjugacy (or similarity) transformation with a unitary matrix U (i.e., In contrast, given Λ ∈ O(1,4), S ∈ Sp(1,1) is determined up to a sign. Still, SO (1,4) where α, β = 1, 2, 3, 5. Γ is a solution of Eqs. (2.8) and (2.9) if and only if it is a linear combination (with complex coefficients) of Γ 0 and Γ 1 Γ 2 Γ 3 Γ 5 . 3 This concludes the construction of Cl s 1,4 . By convention, we use Γ and its inverse to lower and raise the indices a and b of Γ Aa b . We also note the identity ΓΓ † A = −Γ A Γ and that (Γ) 2 ∝ 1.
We denote the 5D metric by h MN , choose the fifth coordinate x 5 along the Killing field and foliate the 5D space-time along x 5 . Since we are interested only in null 5D propagation, we set the lapse of the foliation to 1 by a conformal transformation and denote the shift of the foliation by N µ . The transformed 5D metric is denoted byh MÑ (3.10) To describe particle propagation, we choose a local 5D Lorentzian frame e A M , at each point of the 5D space-time, which we call fünfbeiñ We further provide Cl s 1,4 structure to the tangent space T p at each point p of the manifold. That is, the fünfbein components of a tangent vector x A at p yield an element x = x A Γ A of a locally defined Cl s 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 fünfbein to reflect this where 0 denotes here a 4 × 1 matrix with null entries, and f α µ is a local Lorentzian frame (vierbein) at each point of the 4D space-time with the metricg µν . We further ask that the 5D manifold be interpreted as a conformally flat 4D space-time with superimposed electromagnetic fields. In this case, the 4D foliated metric isg µν = η µν and we choose f α µ to be the Kronecker-delta tensor; i.e., f α µ = 1 α µ . Furthermore, in cgs units, we have N µ = −qA µ /c 2 , where q is specific charge, A µ is the electromagnetic potential, and c is the speed of light in vacuum.
We describe 5D null propagation by assigning two 5D null vectors (or their representations), one ingoing 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 null vector 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 Lie group. Hence, we represent the pair of 5D null vectors as a biquaternion, 6 and describe 5D null propagation by a map Ψ : M → H 1,1 . The biquaternion Ψ(x M ) at point p, with coordinates x M , transforms with elements of Sp(1,1), which is simply connected and compact.
Next, we require the map Ψ(x A ) be Clifford differentiable [21] (i.e., differentiable with respect to x = x A Γ A ) in every tangent space endowed with Cl s 1,4 structure. This requirement constrains Ψ(x A ) to satisfy the following differential equation The square norm of a Clifford element x is given by the product inducing a 5D Riemannian metric. 5 In the 4D picture, this would imply smooth transformations between particle and antiparticle states. 6 A pair of 5D null vectors has the same number of real degrees of freedom as a biquaternion.
As it turns out, the constraint (3.13) is already sufficient for describing 5D particle propagation. We are particularly interested how Ψ(x A ), solution of (3.13), behaves on the local null cone We may write the derivative in the local tangent space as ∂ A = e A M ∂ M and cast Eq. (3.13) over an open neighborhood of the 5D manifold This completes our description of 5D null propagation, as an alternative to the null path-integral formulation [15][16][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 metricsh MN that are independent of both x 5 and x 0 . Hence, we derive 5D equations for quaternionic wavefunctions, starting from Eq. (3.14) rewritten as In the primed representation for the Γ matrices (Appendix A), we obtain where σ j are the Pauli matrices and Ψ 1,2 are the quaternionic components of Ψ. We impose 7 ∂ 0 N α = 0 and ∂ 5 N α = 0, and apply the operator σ j (∂ j + N j ∂ 5 ) on the left of Eqs. (3.16) and (3.17), mindful of the commutator It is interesting to contrast Eq. (3.19) to the 5D Klein-Gordon (KG) equation [15] ▽ where▽ M is the covariant derivative of the metrich MN (3.10) and only ∂ 5 N µ = 0 is imposed. The operator 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 p A in flat 5D space-time (i.e., a 5D photon), we impose |p j | ≪ |p 0 | and |p j | ≪ |p 5 |, which yield p 0 ≈ ±p 5 . Hence, a particle starting at the origin of the coordinate frame remains localized around x 0 ≈ ±x 5 . The propagation problem reduces to the 5D KG equation, which, in light-cone coordinates, leads to the Schrödinger and the Fokker-Planck (FP) equations [15]. In the case of weak fields, Eq. (3.21) still provides approximate light-cone coordinates where The operators ∂ 0 and ∂ 5 in the RHS of Eqs. (3.19) and (3.22) may be explicitly written as subject to further approximations, depending on the application. 7 In this case, the Lorentz and Coulomb gauges are equivalent.

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 where ψ is a bispinor. Using Eq. (3.12), left-multiplying by i Γ 5 , and performing a Fourier transform with respect to x 5 (use that x 5 is Fourier conjugated to λ −1 = mc/ ), we obtain where the matrices γ α ≡ Γ 5 Γ α form the basis for a foliated 4D Clifford algebra where and we use hatˆto denote the Fourier transforms of 5D wavefunctions with respect to x 5 . 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 x 5 . In general, this constraint depends on the electromagnetic field. 8 If the 4D foliated metric isg µν = η µν and f α µ = 1 α µ , then Eq. (4.25) yields the Dirac equation A gauge transformation of the electromagnetic field may be introduced in Eq. (3.14) through a 5D transformation of coordinates: If the transformation is passive, it applies to the coordinates on the 5D manifold, resulting in a transformation for the shift of the foliation N µ → N µ + ∂ µ N , which satisfies the Maxwell equations [16]. If the transformation is active, then one changes (4.28) A Fourier transform of Eq. (4.28) with respect to x 5 yieldŝ where A(x µ ) is the gauge function of the electromagnetic field. Hence, as expected, the electromagnetic field can be interpreted as a U(1)-gauge field for 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][25][26][27]. Still, the Dirac equation has been criticized for the Klein paradox [28][29][30][31] and causality violation [32][33][34][35] in quantum mechanics. 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 not only allows for revisiting the foundations of the Dirac equation, but also brings additional concepts into play. First, the particle (passive) mass appears through Fourier transform with respect to x 5 . Second, electromagnetic 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.

On the non-relativistic limit of the Dirac equation
To describe the non-relativistic propagation of a quantum particle (i.e., p 5 > 0) in time-independent, weak fields, we use y 5 ≈ x 5 and p 0 ≈ p 5 (i.e., ∂ 0 Ψ ± ≈ ∂ 5 Ψ ± ) in the terms containing fields in Eqs. (3.19) and (3.22) to obtain We perform a Fourier transform of Eq. (4.30) with respect to x 5 and use y 0 ≡ cτ as time coordinate, conjugated with the kinetic energy, only. In typical notation of 3D physics, this is where B = ∇ × A and E = −∇A 0 are time-independent magnetic and electric fields, and e ≡ qm and µ ≡ q /(2c) are, respectively, the particle's charge and magneton. The wavefunctionΨ ± may be cast into a spinor pair (Appendix B) for a non-relativistic picture of spin 1/2 quantum mechanics. Equation (4.31) represents the Pauli equation, except for the term 9 ∓iµ σ EΨ ± , describing an interaction between spin and time-independent electric field. Lately, many spin-electric interactions have been studied, both theoretically and experimentally [36][37][38][39][40][41]. However, in contrast to previous work, our spin-interaction term is anti-hermitian, conventionally describing decoherence, decay and resonant scattering induced by electric fields on spin 1/2 particles. From the perspective of non-hermitian quantum mechanics, Eq. (4.31) poses thus an interesting problem. In this case, it is possible to combine particle-spin states of Eq. (4.31) with complementary antiparticle-spin states, to form bispinors described by the Dirac equation. 10 Since the Dirac operator is essential self-adjoint [42][43][44][45][46], decoherence, decay and resonant scattering may be avoided in the phenomenology of relativistic particleantiparticle wavefunctions.
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 where H is the hamiltonian operator acting on spinorsψ. We analyze Eq. (4.32) using the quantum theory of decay processes [47], but other approaches [48] may apply, as well. The probability 11 that stateψ, prepared at τ = 0, remains unaltered is given by (4.33) Thus, within an ensemble of identical particles, the number N (τ ) of particles in stateψ at time τ is N (τ ) = N (0)Pψ(τ ). If Pψ(τ ) decreases with τ , then we say thatψ undergoes spontaneous decay. If Pψ(τ ) increases with τ , then we say thatψ gets populated. In our case, we have exp(−iHτ / ) = cos(ωτ ) + i σ ω ω sin(ωτ ), (4.34) where ω ≡ µ( B + i E)/ and ω ≡ ( ω) 2 . When E is a small perturbation to spin precession in magnetic field B, the first order expansion 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 θ τ / ). (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|), which determines the coherence time and spectral width of the transition between spinor states prepared with the field B. This may be checked experimentally using magnetic resonance spectroscopy.

On the 5D Dirac equation in statistical physics
Quantum propagation has been previously invoked for a space-time picture of statistical physics [49]. Here we use Eq. (3.14) to propose a single-particle formalism for spin 1/2, in line with earlier work for the spinless case [15,16]. This requires a 5D space-time with time-independent metric. Hence, we imposeh MN (3.10) be independent of both x 5 and x 0 . For connecting with previous results and mathematical convenience, we analyze non-relativistic dynamics in the presence of weak fields. Combining Eqs. (3.19) and (3.22), we obtain We introduce spinor rather than quaternionic wavefunctions, perform a Laplace transform with respect to y 0 , and give y 5 ≡ cu the meaning of time. Thus, searching for solutions in the form ψ ± (y 0 , x j , u) = exp(−y 0 /Λ)ψ ′ ± (x j , u) yields equations for ψ ′ ± (x j , u), where evolution is described by the dilatation group D(1), rather than U(1). Spinor eigenstates of these equations can be used to construct the non-relativistic bispinor propagator from ( x, 0) to ( x, u), denoted by ψ ′ ( x, 0; x, u). In turn, the trace of this propagator over the space and spin degrees of freedom provides the Massieu function Ψ/k B for a single particle at constant temperature Ψ/k B = ln d x Tr ψ ′ ( x, 0; x, u). (5.37) In Eq. (5.37), u becomes a time-scale parameter called quantum of physical time [16], to be determined according to the corresponding thermodynamics. A major limitation of the above formalism 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; here, it remains unaccounted for. While the role of temperature for the propagation problem defined by Eqs. (5.36) and (5.37) 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.

The free particle
For the flat 5D metric, the spin degrees of freedom may be ignored and Eq. (5.36) reduces to an equation for a scalar function ψ. A Laplace transform with respect to y 0 yields 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 ≡ M c/ ), with the advantage of more flexibility in adding fields to Eq. (5.38). The , are eigenstates of ∇ 2 in Eq. (5.38). Thus, we may write and We choose u = 2m(βD) = 2m/ζ [16], such that e v u/ = βm v 2 /2 and Ψ/k B yields the traditional thermodynamics for an ideal gas consisting of a single particle.

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(−T D /T ) to explicitly request particle diffusion be negligible; i.e., T ≪ T D . Still, thermal energy may be stored by the spin degrees of freedom. Under these conditions, Eq. (5.36) reduces to Hence, we recover Eq. (4.32), where we now require D(1) evolution. The hamiltonian h = −µ σ(i B ∓ 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. (5.42) may be written as where Ω ± ≡ µ(−i B ± E)/ and Ω ± ≡ ( Ω ± ) 2 ≈ µ(| E| ∓ i| B| cos θ)/ . Taking the trace over spin and spinor degrees of freedom, we obtain Hence, the electric and magnetic susceptibilities of the spin 1/2 particle are given by respectively; the case | E| = 0 is formally recovered in Eq. (5.46) by substituting cos 2 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 requested T ≪ T D .
The above results may be important for current discussions on superconductivity theory [50,51]. Conceivably, the superconduction mechanism may rely on interactions between spins aligned by an external electric field.

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 mechanics is that the 5D metric be independent of x 0 [16]. Strictly speaking, it is not necessary that the 5D metric be independent of x 5 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 the x 0 and consider the role played by the shift of this foliation in statistical mechanics. These fields, included in h 0M , 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 Two representations for the Γ matrices
Sp(1,1) is a pseudo-simplectic group whose generators can be represented in the following form 14 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), subgroup of Sp(1,1), which is a double cover of O(4). It is straightforward to write four Γ-matrices that generate the corresponding Cl 4 Clifford subalgebra where σ j are the Pauli matrices (i, j, k, ... = 1, 2, 3) and iσ j 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 satisfy Γ αa b Γ βb c + Γ βa b Γ αb c = 2(1 αβ 1 a 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 commutators of Γ matrices We now introduce Direct calculations provide Γ Aa b Γ Bb c + Γ Ba b Γ Ab c = 2η AB 1 a c , and the following commutation relations For applications, we note that the matrices γ α ≡ Γ 5 Γ α , given by form the basis for a 4D Clifford algebra where γ αa b γ βb c + γ βa b γ αb c = −2η αβ 1 a 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 = U Γ A U −1 with the unitary matrix We obtain The matrices γ ′α ≡ Γ ′5 Γ ′α are given by 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.61) by changing the sign of γ ′0 ; i.e., γ ′0 → −γ ′0 .