Feynman Path Integrals

The Feynman path integral formulation of quantum mechanics is a powerful and elegant way to describe the dynamics of quantum systems. Previously, we have seen the state vector formalism of quantum mechanics, which is based on a few postulates. The path integral formulation is based on the idea that the probability amplitude for a particle to go from one point to another is given by a sum over all possible paths that the particle can take. This is a very different way of thinking about quantum mechanics, and it leads to many interesting results.

Table of Contents

• Table of Contents
• Propagators
• Evaluating Path Integrals
  • Gaussian Integrals
  • Lattice Discretization
  • Stationary Phase Approximation
• The Three Approaches
• Physical Interpretations
  • Propagators as Green's Functions
• Summary and Next Steps

Propagators

We first consider the matrix elements of the time-evolution operator , which is defined as:

which is to be interpreted as the probability amplitude for a particle to go from point at time to point at time . More specifically, it is a transition amplitude between two states, which is a function of the initial and final states. This is known as the propagator. To see why this is important, consider the position-space wavefunction , which is defined as:

Now we can insert a completeness relation with another set of states :

Hence, the propagator is a kernel that allows us to evolve the wavefunction in time.

To evaluate the propagator, the key is to split the time interval into small intervals of size , where . To that end, beginning with , we can first split the exponential into factors:

Next, we insert complete relations of between each factor of the exponential:

For notation convenience, let's relabel as and as :

We can examine each term in the integrand; we have a general expression (a mini-propagator) of the form:

where the indices run from to . The propagator is then

First, plug in :

Next, we need to use the Trotter product formula to split the exponential into two parts. Recall that the Trotter product formula states that for any two operators and ,

For a proof, see the appendix. Because we are taking the limit or , we can ignore the higher-order terms. Applying this back into the mini-propagator, we have:

It would be nice if we could replace the operators and with their eigenvalues, but although is an eigenstate of , none of the states are eigenstates of . As such, we can insert a completeness relation of the momentum eigenstates :

The inner product of the position and momentum eigenstates is given by:

For the second term (matrix element), because all the exponentials are Hermitian, we can freely extract them out of the inner product. We can apply the exponential to the momentum eigenstate and the exponential to the position eigenstate .

We also get another inner product of the momentum and position eigenstates. Plugging that in, we have

with a negative sign in front of the last exponential because the inner product is conjugate symmetric. This is just a Gaussian integral. Recall . In our case we have and .

With this, we can finally plug this back into the propagator

and take the limit , :

We can expand the product on each term:

A product of exponentials is just the exponential of the sum, so we can instead write this as

As the limit is taken, we have , where is the velocity of the particle (this is just the definition of velocity). We also have . Thus, we can write the propagator as

Miraculously, we have the action in the exponent. The shorthand notation for the propagator is

This is known as a functional integral because instead of integrating over a range of values, we are integrating over all possible functions . The notation is the integration measure, defined as

warning

Some sources try to state that in the path integral formulation, particles explore "all possible paths" between two points. This is technically not true due to a subtle issue—these paths are not paths in physical space, but rather in the abstract configuration space of the system. If, for instance, we have two particles, the configuration space is 6-dimensional while the physical space still has 3 spatial dimensions. This will be crucial when discussing historical attempts to make quantum mechanics relativistic.

Evaluating Path Integrals

If it is not obvious, the path integral is a very difficult object to evaluate. It is perhaps the most difficult object in all of quantum mechanics.

Gaussian Integrals

As vaguely mentioned before, Gaussian integrals are involved in the path integral. We already know that

where is a positive real number. Now let's try to generalize this to integrating over vectors. Suppose we have an -dimensional vector , and we want to integrate over all of . Let's consider the integral

where is a real, symmetric matrix. When compared to the Gaussian integral, we have just replaced scalars with vectors and matrices. First, we consider a similarity transformation , where is an orthogonal matrix and is a diagonal matrix with the eigenvalues of on the diagonal:

Next, we can change variables to . When we integrate over volumes, we need to consider how much the volume element changes as we change variables. For example, if we have and we change coordinates to , then the volume element changes to . The amount by which the volume element changes is given by the determinant of the Jacobian matrix. But we have as an orthogonal matrix, so the determinant is just . Thus, we have

As is diagonal, :

Now we just have an ordinary Gaussian integral in the product:

When we sum over the elements of a diagonal matrix, that is just the determinant:

and the determinant of is the same as the determinant of because it is invariant under similarity transformations.

Next, suppose we add a linear term to the exponent:

where is a vector. This is similar to a shifted Gaussian integral:

When we derive this result, we have to complete the square in the exponent. This involves finding the minimum of the polynomial . Likewise, we can follow the same procedure for the vector case.

Define the polynomial as

To find its minimum, we set the derivative of to zero:

where we have used the fact that . This is because if , then , and if , then since coordinates are independent, . (Yes, it is better to use tensor notation here, but we have not yet introduced it). Anyways, the solution to this equation is

Plugging this back into the polynomial, we have

The reason we are doing this is because now we can complete the square in the exponent. Recall that in normal Gaussian integrals, we find . This helps us rewrite the integral as

from which we just make the substitution . Likewise, let's find :

Now, we make the substitution . This gives us

As one can see, we just have a prefactor of .

Why are we doing this? Consider the fact that functions can be represented as an infinite-dimensional vector space. We can represent the function as a vector , where the components of the vector are the values of the function at each point in space. So our next goal is to evaluate the path integral

We just convert this to vector form, evaluate it, and then convert it back to functional form:

There are a few things to note about this result.

1. The prefactor is the same as the Gaussian integral, but now we have a functional determinant instead of a matrix determinant.

2. The inverse of a multivariable function is a bit more nuanced to define. For a single-variabled function, we just have such that . For a multivariable function, we have to define the inverse in terms of the Dirac delta:

   In other words, the inverse of is just the Green's function of the operator .

Lattice Discretization

Another approach to evaluating path integrals is to discretize the path integral into a lattice. In this approach, we divide the time interval into small intervals of size , where the position of the particle at each time step is represented by . This is essentially what we had in the derivation, before we took the limit .

First, consider the action

where is the Lagrangian of the system. Recall that integrals can be approximated by Riemann sums (for a simple function, it can be visualized as a series of rectangles under the curve). We can approximate the action as

where and . This gives us the discretized action

Recall that we define the path integral as

Removing the limit of in the measure and plugging in the discretized action, we have

This is a discretized version of the path integral, where we have replaced the functional integral with a product of ordinary integrals over the position at each time step. The key is that instead of a functional integral, we have a well-defined, albeit very high-dimensional, integral over the position at each time step.

We will see that this approach is useful for numerical simulations of quantum systems, such as in quantum Monte Carlo methods. This technique is especially important for lattice gauge theories (such as quantum chromodynamics), where the path integral is evaluated on a discrete spacetime lattice.

Stationary Phase Approximation

Another way to approximate path integrals is to use the stationary phase approximation. The key is that for some systems, we already know what the classical path is, and we can use that to approximate the path integral. The stationary phase approximation is based on the idea that the main contribution to the path integral comes from paths where the action is stationary, i.e., where the action does not change much as we vary the path.

Let the classical path of the system be , which is the path that extremizes the action. Each path can then be written as a perturbation around the classical path as

The action can then be written as a Taylor expansion around the classical path. This gives us

The Three Approaches

We have now seen three different approaches to analyzing time evolution in quantum mechanics:

1. The Schrödinger picture in the state vector formalism, where we have the time-evolution operator acting on the state vector . The equation of motion is given by the Schrödinger equation:

2. The Heisenberg picture in the state vector formalism, where we have the time-evolution operator acting on the operators . The equation of motion is given by the Heisenberg equation of motion:

3. The path integral formulation, where we have the propagator defined as the matrix element of the time-evolution operator . The equation of motion is given by the path integral:

Currently, it may seem like the path integral formulation is just far more complicated than the other two approaches. This is the case for simple systems, but path integrals are extensively used in quantum field theory and statistical mechanics.

Physical Interpretations

If one looks at the path integrals, notice that we have the integrand as a complex exponential. The complex exponential is a periodic function that oscillates over time. This means that the path integral is a sum of oscillating functions, which can lead to interference effects.

This is similar to wave behavior in classical physics, where waves can interfere with each other to create constructive or destructive interference. In order to derive an expression for the intensity of light given a wave, we have to take all the possible paths that light can take to get from point A to point B. For each path, we associate a phase factor, then sum over all the paths. If the phases cancel out, we get destructive interference, and if they add up, we get constructive interference.

Similarly, in the path integral formulation, we can think of the paths as being weighted by a phase factor of . Here's the key insight: since there are infinitely many paths, most of them will cancel out with one another. However, in paths where the phase factors are very close to each other, they will add up constructively. This provides a theoretical basis for the idea that particles travel along paths that have a stationary classical action. This is known as the principle of stationary action, which is to be fully discussed in the chapter on field theories.

Propagators as Green's Functions

Taking the definition of the propagator in Equation and inserting a completeness relation of an arbitrary complete set of states , we can express the propagator as

First, notice that the term is just the wave function of the state at position and time . Thus, with and fixed, the propagator is another wave function, and therefore it obeys the Schrödinger's time-dependent wave equation

Second, consider what happens as . In this case, the propagator in becomes

These two properties of the propagator, that it is a solution to the Schrödinger equation and that it reduces to the Dirac delta function as , make it a Green's function of the Schrödinger equation. In other words, it obeys

Summary and Next Steps

The path integral formulation of quantum mechanics is a powerful and elegant way to describe the dynamics of quantum systems.

Here are the key points to remember:

• The propagator is the probability amplitude for a particle to go from point at time to point at time . It is the matrix element of the time-evolution operator .

• Through a series of steps, we can express the propagator as

  Equivalently, we write a shorthand notation for the propagator as

  where is the action of the system and is the integration measure, defined as

• We can solve one class of path integrals, inspired by the Gaussian integral, by completing the square in the exponent:

  where is a real, symmetric matrix and is the inverse of the operator , defined such that

• Two other approaches to evaluating path integrals are the lattice discretization and the stationary phase approximation.

  • The lattice discretization involves dividing the time interval into small intervals and approximating the action as a sum over these intervals.
  • The stationary phase approximation is based on the idea that the main contribution to the path integral comes from paths where the action is stationary, i.e., where the action does not change much as we vary the path.

• The path integral formulation is a powerful tool in quantum field theory and statistical mechanics, allowing us to analyze complex systems and derive results that are difficult to obtain using other methods.

• Path integrals provide a natural way to incorporate the principle of stationary action, where paths that contribute significantly to the integral are those for which the action is stationary.

In the next section, we will finally discuss our first system, the quantum harmonic oscillator.