Quantization of Fields by Averaging Classical Evolution Equations

This paper extends the formalism for quantizing field theories via a microcanonical quantum field theory and Hamilton's principle to classical evolution equations. These are based on the well-known correspondence under a Wick rotation between quantum field theories and 4-D statistical mechanical theories. By placing quantum field theories on a 4+1-D under Wick rotation to 5-D, expectations of observables are calculated for a microcanonical field theory averaging Hamiltonian flow over a fifth spacelike dimension, a technique common in lattice gauge simulations but not in perturbation theory. In a novel demonstration, averaging pairs of external lines in the classical Feynman diagrams over the fifth dimension generates diagrams with loops and vacuum fluctuations identical to Standard Model diagrams. Because it is microcanonical, this approach, while equivalent for standard quantum fields theories in the Standard Model, is able to quantize theories that have no canonical quantization. It is also unique in representing expectations as averages over solutions to an ordinary, classical PDE rather than a path integral or operator based approaches. Hence, this approach draws a clear connection between quantum field theory and classical field theory in higher dimensions which has implications towards how quantum effects are interpreted. In particular, it raises questions about how violations of the ergodic hypothesis could influence quantum measurements even in standard, non-statistical quantum field theory.


I. INTRODUCTION
Microcanonical quantum field theory first appears in the literature in the early 1980's.
Strominger developed a perturbation theory for the φ 4 theory in the microcanonical ensemble hoping to extend it to quantum gravity [1]. Iwazaki later proved the existence of the perturbation series [2] and applied it to fermions [3]. Creutz developed his demon-algorithm based microcanonical quantum field theory around the same time [4]. Meanwhile, in a pair of seminal papers, Callaway developed microcanonical quantum field theory for lattice gauge computations [5] [6]. In his method, Callaway proposes that discrete fields such as the electromagnetic vector potential, φ n,µ , have conjugate momenta with respect to a second time dimension τ , p n,µ = ∂φ n,µ /∂τ . He points out that in the canonical quantization any quantity independent of the field φ can be added to the action. Thus, an observable expectation with the hybrid Monte Carlo or molecular dynamics approach to lattice gauge simulation [7].
and is a unique function of E.
The equivalence of equations 2 and 3 follows from what Callaway refers to as the "principle of equal weight" which states that given E, trajectories given by solutions to Hamilton's equations cover the fixed energy hypersurface with equal density. That expectation values can be computed in either ensemble is discussed in [5] and references cited therein. This principle follows from the ergodic hypothesis. As long as the initial conditions are chosen appropriately and the system has sufficient mixing, all points on the energy hypersurface will be visited with equal probability over infinite τ . In this case, a time average is equivalent to an average over all phase space.
Strominger and Iwazaki present continuum microcanonical quantum field theories as limits of discretized lattice theories. Likewise, Callaway's discrete Hamiltonian system can be extended to continuum field theory by taking the limit of the number of lattice sites to infinity, N → ∞, and the lattice spacings to zero. Hamilton's principle can then be applied via the Euler-Lagrange equations in the standard field theoretic way, where L is the Lagrangian L = T − V . In this case, we derive a partial differential equation indexed by coordinates x rather than a set of ordinary differential equations indexed by lattice sites n. Provided that L is not directly dependent on τ , energy H = E is conserved by these equations. This provides a starting point for a perturbation theory or a computational solution to the PDE.

II. METHOD
We begin by assuming that all field theories exist on a 4+1-D flat manifold, such that the usual spacelike coordinates, x, y, z, are joined by a fourth spacelike dimension w and the metric has signature (− + + + +). We could take w to be a timelike dimension as well or follow [8] and leave it undetermined. In the case of a scalar theory it makes little difference, but, for other theories of interest, particularly gravity, this may create problems such as negative cosmological constants. More immediately, it is easier to present fermions on a de Sitter rather than anti-de Sitter spacetime [9]. We use capital letters A, B, C as 5-vector and 5-tensor subscripts, numbered 0 through 4 with x 4 = w. Small Greek letters, µ, ν, λ, represent 4-vector and 4-tensors, numbered 0 through 3. All fields, ψ, φ, A B , g AB , . . . , are parameterized by this space-like dimension, w. At every slice w is a single instance or microstate of the classical 3+1-D universe. All vectors and tensors are assumed to have indexes from 0 through 4 (five indexes). We show below how assumptions on initial conditions, choice of gauge, and averaging can remove the additional index. Under Wick rotation t → iσ, all fields are in equilibrium in the w dimension; hence, actions are integrals over the four usual dimensions, x µ = (t, x, y, z), which we write as S = d 4 xL.
We will work in double Wick rotated space to put the equations in more familiar statistical territory: t → iσ and w → iτ , making τ timelike and σ spacelike. This has the benefit that reversing the Wick rotation of time gives us quantum field theoretic results while having the additional dimension timelike makes the calculations look more natural (wave rather than Poisson equations) while not affecting the results. This means the signature becomes Given any standard action, S, over a field φ and coupling g 0 the corresponding Hamiltonian is the kinetic added to a potential energy, V , where S = βV , β = 1/g 2 0 , and T = 1 2 d 4 x|p| 2 where p = ∂φ/∂τ . By a Legendre transform, we also obtain a Lagrangian, If the field is a vector A µ , then T = 1 2 d 4 xp µ p µ and likewise for higher order tensors.
The φ 4 Lagrangian is well-known . By the Euler-Lagrange equations, or

III. PERTURBATION THEORY
In this section, the derivation the perturbation theory is given for the scalar field. Let This equation has a perturbation solution.
We are interested in monomial potentials. Let u(x, τ ) be a nonlinear potential such that our KG equation is, Applying Green's function method to solve the nonhomogeneous equation, where the propagator (Green's function) is, and k 2 = k 2 0 + k 2 1 + k 2 2 + k 2 3 , and kx = k µ x µ . Now, we want to solve 7 when 0 < λ m 2 . This method is straightforward perturbation theory [10]. Let the solution be the perturbation series, When we plug this into the equation we get, Collecting coefficients we find, Because k, l, m < n, this creates an iterative solution. Starting with the free solution, given by a sum of plane waves: where The position space free solution is φ 0 (x, τ ) = d 4 k/(2π) 4 e ikxφ 0 (k, τ ). We then use that solution to compute the next solution, then that for the following solution, which allows our previous solution to substitute for φ 1 , and so on so that all solutions can be found in terms of interactions of the free solution φ 0 (9). The integrals over χ and χ indicate interactions between different slices of τ , the classical 3+1-D universes.
Let the perturbation solution be truncated to level N , such that and discard all solutions of order O(λ n+1 ). (In general, the solution will diverge as N → ∞ as in standard perturbation theory.) The 2M -correlation Green's functions against the vacuum Ω in momentum space are, Therefore, the expected value of the correlation of two plane wave solutions averaged over τ is, Let A(k) = √ 2, which gives the equivalent correlation to standard path integral quantization when = c = 1. This gives the initial condition for φ as well (an amplitude scaling that turns out to be constant for all k). To regularize this for renormalization, which we will need to do, we will let A(k) fall to zero at some large |k| = Λ, but for now we will let it remain constant.
The momentum space version of one of our solutions (keeping it in terms of τ ), Taking the mean gives the propagatorD(k) = D (k; τ ) = 1 k 2 +m 2 . Taking the negative gives the usual propagator after reverse Wick rotation. The delta function ensures conservation of momentum in the interaction. Higher order perturbations can be found similarly.

IV. FEYNMAN RULES
Suppose we want the Green's function in momentum space up to order λ, Carrying out the perturbative calculation, the correlation has three terms. The first, a zeroth order correlation, involves two free fields and no interaction. The second and third involve correlation between a free field and a single interaction.
These terms can be laboriously computed but there is a simpler approach with Feynman diagrams. Let the propagator be the field φ g = −D(x − y; τ − χ). The solution for position This represents bringing three plane wave solutions, φ 0 , to a single point y and transporting that to x using the Green's function, φ g . This pattern can be iterated to higher orders adding more points that transport more particles together with symmetry factors representing the permutations of such graphs.
Helling [11] writes down the rules for these classical Feynman diagrams as follows: 1. Draw n vertices for the expression for φ n at order λ n .
2. Each vertex gets one in-going line at the left and three outgoing lines to the right. These rules are nearly identical to the Feynman rules for the equivalent scalar statistical theory with the only difference being that loops are prohibited because of rules #2 and #3 on incoming and outgoing lines. Relaxing these rules allows for loops.
When the expected value is taken by averaging over τ , we find that the correlation of pairs of plane wave solutions and the propagator, which are different in the 4+1-D evolution theory, become equal in the statistical theory. Reverse Wick rotating back to real time: In real time, a factor of i is introduced in the example in the preceding paragraph: . The diagrammatic equation correlates the first order diagram with a single, zeroth order plane wave solution (just a line), The correlation also need not be between a diagram and an external line. As seen in the next section on QED, two complete diagrams can be correlated.

V. QUANTUM ELECTRODYNAMICS
Following [11] and [6], these scalar field results can be extended to other quantum theories, adding a kinetic energy term in τ and solving the resulting PDE perturbatively, where the same transformation from pair correlation to propagator occurs in the τ average. In quantum electrodynamics, there is a cubic term in the Lagrangian, These produce the equations of motion, and assuming the Lorentz gauge condition, ∂ µ A µ = 0.
Add another spatial dimension, x 4 = w, a w dependency to the 5-vector field A B and spinor field ψ, and, in addition to the Lorentz condition, require the gauge condition A 4 = 0.

Now double Wick rotate, and the classical 4+1-D equation has the form,
This equation can be extended with classical ghost fields [12] as well, but these do not become essential until the local gauge symmetry of non-Abelian theories demands them.
The extended Dirac Lagrangian has an additional term −ψγ 5ψ which couples the spinor toψ = ∂ 4 ψ so that it cannot be, e.g., normalized out of the path integral.
It is possible, however, to use the de Sitter space Dirac equation and still achieve standard results. As far back as 1935 in his paper on the electron wave equation in de Sitter space [9], Dirac wrote down an alternative but equivalent form for the Dirac equation in 3+1-D, where κ is a mass term [13].
The perturbation theory now represents the interaction of the two fields. Let A µ = ∞ n=0 A µ,n e n and ψ = ∞ n=0 ψ n e n . The calculations are similar to the scalar theory and have the same result of reproducing quantum field theories in 3+1-D. Because the wave equation 19 has the right hand side, eψγ µ ψ, which is quadratic in the fermion, we have diagrams with two fermions on the right connecting to one photon propagator on the left where one fermion is always going in and one always coming out. The Dirac equation 20 is bilinear in the photon and fermion. Thus, it connects one photon to one fermion on the right and one fermion propagator on the left. These form tree structures as pictured in [11].
As shown below, using the power of averaging, diagrams can be correlated together to form the more complex diagrams of quantum field theory.
In the following we show that common scattering results such as fermion scattering ψ (x 1 )γ µ ψ(x 1 )ψ(x 2 )γ ν ψ(x 2 )A g,µν (x 1 − x 2 ) and Compton scattering ψ (x 1 )γ µ ψ g (x 1 − x 2 )γ ν ψ(x 2 )A µ (x 1 )A ν (x 2 ) agree with those of standard quantizations. The correspondence of pair correlations of free, plane wave solutions and propagators in the average over τ , The relation ψ(x, τ )ψ(y, τ ) is an outer product of two four component spinors which, when composed of a pair of plane wave solutions, produces a matrix, which, in the average over τ , becomes the Dirac propagator in 3+1-D. Although the outer product to propagator equivalence can be found in many textbooks (e.g., [14]) for 3+1-D, adding the additional dimension requires a slightly different derivation shown in the following: The Dirac equation has four plane wave solutions, the four combinations of spin-up, spindown, positive energy (electrons), and "negative" energy (positrons). This is true in both 3+1-D and 4+1-D. For the following, we will work in real time and space assuming we can do the same in double Wick rotated spacetime. Consider the 5-momentum p A where p t = ±E (with c = 1), (p 1 , p 2 , p 3 ) = p = (p x , p y , p z ) and the final momentum is p 4 = p w . For this we will describe solutions in terms of the Pauli matrices, In the Dirac basis, By the anzatz, ψ = u(p µ )e −ip A x A , u(p µ ) is four component spinor independent of position, x A . Let m = 0 for now. We will reintroduce it later. The Dirac equation in 4+1-D is now, where p 0 = E.
Let u T = (u 1 , u 2 ) with u i being 2-component spinors. The Dirac equation now reads Each of these equations implies the other. In the following, we can use the identity which follows from direct calculation [14]. Negative energy solutions, v 1 and v 2 can be found likewise and these satisfy the completeness relation: where A is constant and ω = p 4 = p 2 − m 2 . Using the same methods and the result 23, one can show that where we have added the small factor of to avoid poles [14]. (We can also show this for the negative energy solutions v s . See pg. 108-9 of [14] for more details.) The additional term in the numerator iγ 5 p 2 − m 2 modifies the mass. On mass shell p 2 = m 2 , however, it vanishes.
The above presents an interesting result. When particles are on mass shell, it means their momentum in the 5th dimension is zero, i.e., they are parallel to that dimension, since if p 2 = m 2 then p 4 = 0. Therefore, a necessary condition for particles to be "real" is that they not be propagating in that dimension, i.e., they are confined to a single microstate or universe.
The diagrams in the classical theory have no loops again, but they develop loops when averaged over τ . For example, using the Feynman rules described in the previous section on scalar theory, modified for QED to include fermion/photon vertexes, the following diagrammatic expression 25 shows the correlation of two second-order classical momentum space diagrams. These have been rotated 45 degrees so the starting fermion propagator is at the bottom. The diagram on the right has been reversed from the convention we have been using so that it starts on the right and moves to the left. This reversal makes it easier to see the transformation of the correlation of the two diagrams into the loop diagram of the quantum field theory. with where f abc are the structure constants of the Lie algebra, For the 4+1-D case [15], it is convenient to choose the gauge A a 4 = 0 and ∂ µ A a µ = 0 (or an R ξ gauge), in which case the free equations are The first equation in 27 is the usual Yang-Mills equation with an "acceleration" term added to it. Thus, it fits within the overall approach from molecular dynamics.
The overcounting problem of the path integral method of quantization is addressed in molecular dynamics by a change of variables from dependent to independent [6]. For perturbation theory, however, the problem requires a continuum approach that addresses all orders of loop renormalization. The problem itself is slightly less intuitive in the evolution equation approach than in the path integral approach where the functional integral is clearly overcounting. Here the overcounting occurs because the PDE is under-determined ( [16] pg. 40-1), a concept Dirac first proposed [17], in its evolution in τ .
This is better understood in the discrete case: for N independent degrees of freedom there are M > N pairs of Hamiltonian ODEs and M functions φ 1 (τ ), . . . , φ M (τ ) (where φ i (τ ) is a discrete element of any field evolving in τ ) [6]. It may appear by analogy with the path integral that the overcounting problem is in the time-average integral, i.e., that information is being counted at τ 1 and later at some τ 2 > τ 1 redundantly counted again so that φ i (τ 1 ) is gauge equivalent to φ i (τ 2 We can resolve the problem using a Faddeev-Popov method of spin-0 anti-commuting ghost fields. These fields remove the redundant dimensions from the PDE solution space. In the Abelian case, equations for the ghost fields are decoupled from the equations for the vector potential. In the non-Abelian case, however, there is no global gauge freedom and so the ghost fields couple to the potential. (A simpler dynamic ghost field can be added by a Bell-Treiman-Veltman transformation instead, but this is only beneficial in one-loop renormalization [18] and some computational schemes where it can condition matrices [12].) The Lagrangian for the ghost fields is [19], wherec and c are scalar fermionic fields, Grassmann numbers. Taking variation with respect toc a , these give the equations of motion, Let the initial condition beȦ a 4 =Ä a 4 = 0 at τ = 0 and, thus,Ȧ a 4 = 0 for all τ . Adding this condition to the choice of gauge, these reduce to, Now both 27 and 29 form the Yang-Mills equations with Faddeev-Popov ghosts with an acceleration term added which agrees with the molecular dynamics approach.
Thus, we have shown that we can reduce the equations to the desired results (no dependencies on the τ dimension other than those internal to the fields which will be averaged and the acceleration term that will drop out in the average). This is necessary in order to achieve equivalence with standard perturbative QFT within the framework of a classical de Sitter spacetime.
Expanding 27 and 29 in terms of powers of g, we can obtain a perturbation series for the classical 4+1-D equations. The approach is almost identical to the scalar case but with more kinds of vertices possible and now coordinate and gauge indexes on the lines. The equation 27 has quadratic terms in the gluons, A a µ , as well as a cubic term. Therefore, following the convention of the previous sections, diagrams begin on the left with a gluon propagator which then leads to vertices that can have two or three gluon lines. These can be plane wave solutions forming external lines or propagators leading to other vertices. The contributions for these vertices turn out to be the same as in the quantum theory. Likewise, the ghost equations have the usual vertices with a vertex that is bilinear with the ghost field and gluon and contributes −gf abc p µ . Averaging over τ has the same result as in the scalar theory of producing the usual propagator as well as allowing different diagrams to be correlated in order to form loops. Fermions can be added in the same way as in QED of the previous section.
Renormalization can be done after averaging (with regularization done before) in which case the approach is the same as in the Standard Model.

VII. CONCLUSION
Discrete Hamiltonian flow method of molecular dynamics has been extended to a continuum PDE method showing that classical Feynman diagrams reduce to quantum field theoretic ones when correlated. By doing so, a novel way of reducing quantum physics on a 3+1-D spacetime to classical physics on a 4+1-D de Sitter spacetime has been presented.
Is this a mathematical trick or physical reality? One way to distinguish the fundamentally (and mysteriously) statistical nature of quantum physics in the standard formulation and the evolution approach presented in this paper is to look at areas where the equivalence breaks down. In particular, the equilibrium assumption allows for averaging over all τ . If this assumption were to break down and the theory to become a non-equilibrium one, the additional dimension might be observable. Another potential mode of detecting is to measure the effect of virtual particles being particles that have momentum in the τ dimension and shown in the section on QED. This suggests that the de Sitter rather than Lorentz invariance may be measured when observing virtual particles that become real and vice versa.
There may be a measureable effect because the velocity of a massless virtual particle in the familiar three dimensions would be less than the speed of light since some of that velocity is in the 4th spatial dimension.