S. Tomonaga's (ST) 1946 paper "On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields" (reprinted in Schwinger's Dover book QED p.156) may provide the answer.

ST gives a four-dimensional form of the field commutation relations. First he defines a "functional" in a footnote. psi[v'(xyz)] means that psi is a functional of the variable function v'(xyz).

By "variable function" ST does not mean that the value of the function varies as its independent variables xyz vary. Rather he means, imagine a space of all possible functions each subject to the same constraints or boundary conditions, perhaps with a "metric" giving the "separation" between two such functions which are "points" in the function space. In general this separation can be a complex number, maybe even a quaternion or an octonion. The functions may not even be continuous and differentiable as in elementary calculus. They may be fractals, or distributions. When ST says "variable" function he means moving from point to point in this function space. A functional is a meta-function. The function idea self-referentially jumps to a level outside itself maintaining its invariant structure. It's a kind of self-iteration.

ST gives an example, the energy density dH(v(xyz) u(xyz))/dxdydz is an ordinary function of xyz, but the total energy

ST considers the problem of two interacting fields 1 and 2. In the non-Lorentz covariant form, the frame-dependent Schrodinger equation is (equations numbered as in ST's paper for reference).

H's refer to total field energies. They are nonlocal quantities defined as integrals over a particular spacelike surface that defines a global inertial frame in flat spacetime. Global inertial frames are not possible in the general relativity of curved spacetime.

ST then goes to an "interaction picture". This is our first problem because Peter Holland says that the Bohm theory is stuck in the Schrodinger picture.

ST defines the unitary operator for the free fields 1 and 2

The Schrodinger picture time-independent fields are u1(xyz) and u2(xyz) with canonically conjugate momenta v1(xyz) and v2(xyz) respectively. Define the unitary transformations

for each field i = = 1 or 2

ST then gives the Lorentz-covariant field commutation relations

"where Aij, Bij and Cij are functions which are combinations of the so-called four-dimensional delta functions and their derivatives."

Consider a 2D surface in 3D wavenumber kx,ky,kz defined by the Lorentz frame-invariant "mass shell" constraint equation

Prove that the element of area on the sphere is dA =(m/k) dkx dky dkz and that it is Lorentz invariant.

The four-dimensional Lorentz-invariant delta functions have the form

Lorentz invariance demands that the first exponential is an advanced plane wave propagating backward in time rather than a retarded wave propagating forward in time opposite to the wave in the second exponential. This because the Lorentz signature is +++- and the invariant scalar product must be of that form, therefore t is a negative quantity in the first exponential and positive in the second. Therefore, the above function is the destructive interference of an advanced wave with its "mirror" retarded wave connecting the two events (xyzt) and (x'y'z't'). (We have set (x'y'z't') at the origin in the above equation (11).)

Equation (11) above obeys microcausality since it vanishes when the two events are spacelike separated.

However, it is possible to define new kinds of causality-violating four-dimensional Lorentz-invariant commutation relations by using the new function D' that has constructive interference of the advanced and retarded waves in addition to the new "tachyonic" mass shell

Note the D' function above is not to be confused with Pauli's "D1" function which still uses the subluminal mass shell. The idea here is that both exotic matter and dark matter require the new causality-violating field theory. The spin-statistics connection will be reversed here so that spin 0 quanta are fermions and spin 1/2 quanta are bosons. This is a prediction for cosmological dark matter of my new theory.

End of Advanced Digression

He generalizes the frame-dependent Schrodinger equation to a Lorentz-invariant equation.

In the interaction picture

t defines a frame-dependent spacelike surface. It is a global time. ST starts with the Dirac (1933) many-time formalism from many-particle quantum mechanics. Dirac studied N charged particles. He first used the unitary operator infinitesimally generated by the Hamiltonian of the free electromagnetic field. Dirac then makes a unitary transformation on the electromagnetic vector potential and on the wave function PSI. In the next step he introduces a new many-time function PHI with as many local time variables as there are particles. Each particle carries its own local clock. The new multiple-time wave function PHI includes timelike separated quantum connections as well as spacelike ones. The result is N simultaneous Schrodinger equations. The key point which involves microcausality is that these N simultaneous equations (eqs 20 in ST's paper) require N^2 conditions

A sufficient condition for a solution is when the N events forming the arguments of PHI are all relatively spacelike separated for all possible pairings if we use microcausality which forbids controllable spacelike actions-at-a-distance. But is this sufficient condition also necessary? My new programe for causality-violating quantum field theory can be wrecked at this stage.

Tomonaga uses the functional derivative to generalize Dirac's discrete finite many-time to continuous infinite number of local times one for each space point. Define the function t(xyz). We then have a functional PHI'[t(xyz)]. Define the variation density &dt(xyz)/dxdydz which differs from zero only in a small region R in the neighborhood of the space point (xyz). Define the functional derivative

Tomonaga's main result is then the Lorentz-invariant Schrodinger functional differential equation for the relativistic quantum field

which can be solved with the microcausal invariant commutation rules.

Thus, our system of equations (26) is integrable when the surface defined by the equations t = t(xyz), considering t(xyz) as a function of x,y and z, is spacelike. ... The dependent variable surface ... can be of any (spacelike) form in the spacetime world, and we need not presuppose any Lorentz frame to define such a surface. Therefore, this PHI'[t(xyz)] is a relativistically invariant concept. ... It is not necessary ... to admit also time-like surfaces for the variable surface as was required by Dirac ..

Any Russians out there on the World Wide Web - take a look and report back please. :-)

The problem now is to make a Hamilton-Jacobi theory for many-times so that Bohm's theory need not be non-Lorentz invariant in the individual case.

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