The path integral method is perhaps the most elegant and powerful of all quantization programs.
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For path integrals we need only a few simple ingredients – wave functions, paths, a Lagrangian – a way to add them up, and a bit of time.
Usually we would start an analysis with the Schrödinger equation and then derive the kernel as its inverse. But here it is more natural to start with the path integral expression for the kernel, and then derive the Schrödinger equation from the kernel. It is a bit like the game Jeopardy, where we are given the answer, the kernel, & have to work out the question, the Schrödinger equation.
The simplest paths are a series of delta functions in coordinates; more generally we can take a path as being as a series of wave functions at successive instants, if we have a good basis to work with.
We are using the usual procedure of slicing the laboratory time from start to finish into successively thinner slices, as a frugal delicatessan owner might, and then taking the limit as N, the number of slices, goes to infinity.
Our rule of treating (quantum) time and space as symmetrically as possible makes most of our decisions straight-forward.
Steps:
To define our wave functions we need a basis which is:
Non-singular.
General.
Reasonably simple.
Using a Fourier decomposition in time is problematic.
The use of singular functions or non-normalizable functions, e.g. plane waves or functions, may introduce artifacts. It is safer to use more physical wave functions, i.e. wave packets.
There is a good example of the benefits of using more realistic wave functions in Gondran and Gondran, [Gondran-2005]. There they argue that some of the problems in interpretation of the Stern-Gerlach experiment are a result of using unphysical wave functions to do the analysis.
Further, use of plane waves as the basis makes demonstrating convergence of path integrals problematic, as will be discussed below.
Consider an incoming beam, say of electrons. Have normally something of the form:
We could generalize the part to:
To make this fully general, note that any square-integrable wave function may be written as a sum over Morlet wavelets [Kaiser-1994].
A Morlet wavelet in one dimension has the form:
Where s is the scale and d is the displacement.
We can decompose any square-integrable wave function into a sum over Morlet wavelets. Analysis:
Synthesis:
C , the admissibility constant, is computed in the Morlet wavelet appendix Morlet wavelets.
As each Morlet wavelet is a sum over two Gaussians, any normalizable function may be decomposed into a sum over Gaussians.
To compute path integral results for an arbitrary normalizable wave function, decompose into Gaussian test functions, compute for each Gaussian test function, then sum to get full result.
For four-space, do decomposition four times.
A Gaussian test function is defined by its values for the average x , average p , and dispersion in x (or alternatively p). For a four dimensional function, there are potentially four plus four plus sixteen or twenty-four of these values. For test functions we will always use a diagonal dispersion matrix, letting write our test functions as direct products of functions in , x , y , and z .
With definition of standard deviation matrix:
Break out into time and space parts. Use for time part; for space.
Time part:
Space part:
Expectations of coordinates:
And uncertainty:
We can use the Gaussian test functions as either "typical" wave forms, to see what the system is likely to do, or as the components (via Morlet wavelet decomposition) of a completely general solution.
A path is a series of wave functions indexed by .
For a particle in coordinate representation, it is a series of functions indexed by , i.e.:
To model a particle, we use narrow Gaussian test functions from above.
Like all paths in path integrals, our paths are going to be jagged, darting around forward and back in time, like a frisky dog being walked by its much slower and more sedate owner. Why we can usually get away with using the smoother path of the owner, rather than the more jagged path of the dog, is discussed below in Semi-classical approximation.
Minimum requirements:
Produces correct classical equations of motion.
Manifestly symmetric between time and space.
Correct non-relativistic limit.
Correct Schrödinger equation.
Reasonably simple.
Per [Goldstein-1980], a Lagrangian of the form
With definition:
Will cause the classical equations of motion to be satisfied.
We may think of the classical trajectory as being like the river running through the center of a valley; the quantum fluctuations as corresponding to the topography of the surrounding valley. Many different topologies of the valley are consistent with the same course for the river.
Therefore the requirement that we get the same classical trajectory does not completely constrain the Lagrangian. In particular, Goldstein notes that we could also look at Lagrangians where the velocity squared term is replaced by a general function of the velocity squared:
Subject to the condition:
So we could explore alternative Lagrangian's, such as:
However our choice for the velocity squared term is the only one which produces the correct classical equations of motion and which is quadratic in the coordinates. It is therefore simplest.
The overall scale and an additive constant are still free.
The scale is constrained by the non-relativistic limit, see below.
And as we will see ( below) if we add a mass term of form:
We will get the Klein-Gordon equation back as our Schrödinger equation.
Candidate Lagrangian therefore:
Break out the Lagrangian in time and space:
This Lagrangian gives the Euler-Lagrange equations of motion:
In terms of electric and magnetic fields:
In manifestly covariant form:
With:
We have the same classical equations of motion in four dimensions as in three, something which does much to explain why temporal quantization has not been seen, see chapter on semi-classical approximation below.
Still see differences at the quantum level, however.
The effect of temporal quantization is to replace the electric potential with three new terms:
Action defined as:
The choice of s involves some subtleties. Per Goldstein, it can be any Lorentz invariant parameter.
The most obvious choice is the proper time of the particle (this was Feynman's choice in [Feynman-1950] [Feynman-1951]). This is a problem, however, in going to a multiple particle system: we now get too many different times, one per particle. This is particularly awkward when there are infinite numbers of virtual particles lurking about. And of course it does not work well for a photon.
We will use Alice's proper time for the time being:
However once we have worked out the rules using this, we will switch to using the geodesic time: see Geodesic definition of laboratory time.
We derive the Hamiltonian from the Lagrangian. Hamiltonian gives insight on Schrödinger equation; and lets us develop alternate form of path integrals.
We have the conjugate momentum to quantum time with respect to laboratory time:
So:
And the conjugate momentum to space with respect to laboratory time:
So:
Hamiltonian given by:
So the Hamiltonian is:
This prefigures the form the Schrödinger equation will take:
The simplicity of this provides support for the choice of Lagrangian above.
Hamiltonian equations for coordinates are:
For momenta are:
These are thinly disguised versions of the Euler-Lagrange equations above.
Can think of the Hamiltonian as corresponding to the laboratory energy , the energy variable conjugate to the laboratory time.
Our path integral measure has to include fluctuations in time as well as the more familiar fluctuations in space. The obvious definition of the measure is:
We start at A finish at B:
Break out the path integral into time slices:
With a normalization constant to be determined below.
Single time step has duration:
With discrete Lagrangian:
The laboratory time functions as a kind of stepper. It is as if we were making a stop motion film, i.e. a Wallace and Gromit feature, and each click forward of by corresponds to a single frame, with Wallace and Gromit being wave functions that interact with each other.
Compare this Lagrangian to the standard quantum theory Lagrangian:
With temporal quantization, the usual potential term becomes more complex:
We multiply the potential by the velocity of with respect to . In the non-relativistic case, this factor will turn into approximately one, giving the usual result back.
The time part of the Lagrangian is new. It has a "kinetic energy" and a "potential energy" term.
The kinetic energy of time is completely new: identical in form to the usual space term, but opposite in sign.
The potential energy, the mass term, is less interesting and if sufficiently annoying may be gauged out of existence, see below.
We are using the mid-point rule: we evaluate the potential at the mid-point between the end times for a slice.
This is already required for evaluations of the vector potential; Schulman points out that failure to use the midpoint rule for the vector potential causes spurious terms to appear in the Schrödinger equation.
Our principle of the most complete symmetry between space and time therefore mandates use of the midpoint rule for the electric potential as well.
Another decision made for us by the requirement of maximally manifest symmetry.
Path integrals usually involve long series of Gaussian integrals, of the general form:
See for instance [Zee-2003] or any quantum electrodynamics text that deals with path integrals.
For these to converge, the real part of a should be greater than zero. However, in most cases, it happens to be exactly zero. Unfortunate.
The traditional response to this problem is to add a small positive real part to a, then let it go to zero. In a path integral context, we will have something like:
And we may add this small real part to either the mass:
Or the time:
There is the obvious question: where did that come from? The candid answer is that it is a magic convergence factor added to make things come out.
Unfortunately, with temporal quantization the magic fails. No matter which sign we choose for this, it cannot be the same sign for both time and space. Looking at a single step in the path integral:
And if we choose different signs for time and space, then we lose manifest symmetry between time and space.
An alternative response is to Wick rotate in time, shifting to an imaginary time coordinate. This has the same problem: no matter in which sense we Wick rotate, either the past or the future integral will be infinite.
We consider several points in turn:
If we limit our wave functions to square integrable functions – which includes all physically meaningful wave functions – then we may write any allowed wave function as a sum of Gaussian test functions.
If we then integrate from our starting slice forward, each integral will in turn be well-defined. The convergence comes naturally from the wave function, not from the kernel.
Therefore, we have no need of artificial means to ensure convergence: it is a natural consequence of restricting our examination to physically meaningful wave functions.
This does mean that the integrals we have to do are more complex than usual: we have to do each integral with respect to a specific incoming wave function, we cannot start in full generality.
And therefore, we will need to pay careful attention to how we normalize the path integrals: the normalization could depend on the specific starting wave function.
The dispersion of the wave function gets larger with time on a per Gaussian test function basis:
So the rate of convergence is different for each Gaussian test function. At each step it is a function of the initial σ and the laboratory time.
However, once the Gaussian test function is picked, convergence is assured.
This means different parts of a general wave function will converge to the final result at different rates.
So long as they do converge, this does not matter.
By relying on Morlet wavelet decomposition, we have avoided magic at the cost of trading unconditional convergence for conditional convergence.
In the usual development of path integrals, normalization is inherited from the Schrödinger equation. Here it has to be supplied by the path integrals themselves.
We will start with one of the normalized Gaussian test functions from above:
Have wave function at end in terms of kernel:
Normalization defined by requirement:
Define unnormalized kernel as the kernel we get from a straight computation of the path integral, with no normalization:
Gives formula for normalization:
Obviously we are free to add an overall phase at each step; gives a new gauge degree of freedom. (Actually this is really an old gauge degree of freedom: see Gauge transformations, also Semi-classical approximation.)
Here we will look at the free case only. Later we will complete the analysis by using the Schrödinger equation to demonstrate unitarity, implying the normalization is correct in the general case.
Separate variables in time and space. work here with first with time part, then generalize to all four.
Gaussian test function:
Kernel:
Expression for wave function after first step:
Explicit wave function after first step:
With definition:
Normalization requirement is:
Single step normalization is correct if we multiply by a factor of:
Making the overall normalization for N + 1 infinitesimal kernels the product of N + 1 of these factors:
As noted, the phase is arbitrary. If we were working the other way, from Schrödinger equation to path integral, the phase would be determined by the Schrödinger equation itself. However, the phase choice we are making here will turn out to be both convenient and conventional.
Expression for kernel is:
Gives for kernel:
And for wave function:
With this normalization we have for the probability distribution in time:
With obvious expectation of time:
Implying velocity in time:
Equal to traditional relativistic factor:
Which is logical.
Uncertainty:
We have done analysis for an arbitrary Gaussian test function; recall that any real function will be a sum over these.
The most important thing about the normalization is what we do not see in it: it is not a function of the frequency, dispersion, or offset of the Gaussian test function. Since any square-integrable wave function may be built up of sums of the Gaussian test functions, the normalization – at least for the free case – is independent of the wave function.
We recapitulate the analysis in time in space.
Use the correspondences:
With these we can write down the equivalent set of results by inspection.
To get for one space factor Gaussian test function:
Free kernel:
As a result, normalization gets a phase change:
Again, phase is arbitrary but convenient.
Kernel becomes:
Explicit value of wave function at later time:
With definition:
Probability distribution:
With expectation of position:
Implying velocity with respect to laboratory time:
Uncertainty:
Full space part is the product of x , y , and z parts; it is in fact the usual non-relativistic free kernel:
Full kernel:
Evolved wave function:
With expectation of coordinate:
And dispersion matrix:
We can get the energy and momentum forms by taking Fourier transform. Free kernel:
And if the initial wave functions are given as above:
With:
Then the free wave function as a function of laboratory time is:
Considerably simpler. While the particle in time/space space expands with laboratory time; in energy momentum space its dispersion is constant.
Both energy momentum and time space forms look different in relative time, see below.
The full kernel is therefore:
With definition of the measure:
This was derived for an arbitrary Gaussian test function, but by Morlet wavelet decomposition is valid for an arbitrary square-integrable wave function.
While we have verified the normalization only for the free case, the verification of unitarity below will imply the normalization is correct in general.