The next few steps involve a small nightmare of Taylor expansions and Gaussian integrals.
Now we are going to go the "wrong" way; we will be running Schulman's ( [Schulman-1981]) derivation of path integral from Schrödinger equation in reverse.
We start with Explicit expression.
Only terms first order in appear in limit.
Constant term gives:
This cancels against normalization factor in the overall kernel.
Odd powers of ξ give zero, off diagonal ξ give zero.
Diagonal terms give:
Expression for wave function therefore:
Or the four dimensional Schrödinger equation:
If we make the customary identifications:
Or:
We have
Or
Using the definition for the Hamiltonian in Lagrangian.
In free case, solutions stationary with respect to laboratory time satisfy the Klein-Gordon equation:
We will argue below that taking the set of stationary solutions picks out the standard quantum theory part of temporal quantization.
If the electric potential is zero we have:
The magnetic potential contributes terms where acts on , as well as on the wave function.
When we have a non-zero electric potential we have similar terms; we have to consider ordering. Time dependent electric potentials will not commute with the energy operator. Therefore write the time/electric potential product terms as:
By precise analogy with the space part.
We also have a term which is the square of the electric potential:
By analogy with the term in squared.
We demonstrate unitarity using the same proof as for Schrödinger equation in standard quantum theory: we form the probability:
Have for change of probability in time:
Schrödinger equation for wave function and its complex conjugate:
Rewrite, throw out cancelling terms, and choose Lorentz gauge.
Integrate by parts; there is nothing left:
Rate of change of probability is zero, as was to be shown.
Normalization therefore correct. Probability conserved.
Unitarity achieved, in four dimensions rather than three.
From a three dimensional point of view, bookkeeping at any one instant approximate. Temporal quantization like a petty cash account; "probability" can hide in time, then return.
We can write the wave function as a product of a gauge function in quantum time, space, and laboratory time and a gauged wave function:
Or:
If the original wave function satisfies the Schrödinger equation, the gauged wave function satisfies the gauged Schrödinger equation:
Provided:
And:
Refer to new potential as Alice's potential. In general:
For example, we can cancel the mass term on the right by choosing a gauge:
When gauge function not a function of laboratory time:
Gauge transformation is familiar:
Or:
In general, will depend on the laboratory time. We can pair a potential with the derivative with respect to laboratory time, just as we pair potentials with the derivatives with respect to quantum time or space:
In fact this is essential for consistency. If we do integration by parts in the Lagrangian, the end terms show up as gauge transformations using a gauge function that is a function of laboratory time.
Gauge transformations important for understanding coordinate changes; see Kleinert and also below Constant electric field.
And useful in constructing approximations, see Time independent electric fields below.
[ 2][ 2]For a more sophisticated treatment of the gauge issues dealt with here see the handling of gauge in fifth parameter formalisms, i.e: [Land-1996] [Horwitz-1998].