Show that the completeness relation is ∑λ εμ^(λ) * εν^(λ)=-gμν+(pμ pν)/(M^2) where the sum is ov

Show that the completeness relation is ∑λ εμ^(λ) *...

Key Concepts

Polarization Vectors

Polarization vectors describe the independent physical states of a spin-1 (vector) particle. For a massive vector particle, these vectors satisfy specific orthogonality and normalization properties, such as transversality with respect to the momentum vector, which are crucial for ensuring that the particle’s description is consistent and only involves physical degrees of freedom.

Metric Tensor

The metric tensor in relativistic field theory is a fundamental object that defines the invariant interval between points in spacetime and is used to raise and lower indices. In completeness relations, the metric tensor appears naturally to maintain Lorentz invariance and to properly project different tensor components in the expression of the propagator or the identity operator.

Completeness Relation

In quantum field theory, the completeness relation states that a complete set of states forms an identity in the subspace they span. When applied to polarization states of a particle, this relation asserts that summing over the outer products of polarization vectors reconstructs the projection operator onto the particle’s physical degrees of freedom.

Projection Operator

A projection operator is built from the polarization vectors and is used to isolate or extract the physical components of a field. In the context of massive vector particles, the sum over the polarization states yields a projection operator that correctly identifies the subspace of physical polarizations by eliminating unphysical degrees of freedom.

Transcript

00:01 So we start with the wave function for a particle in a box, and we'll assume that it is a sign dependence, basically a sign function.

00:11 And we're going to calculate the quantities that are needed for the uncertainties in the uncertainty principle, namely the mean or expectation values of x, x, x squared, p, and p squared.

00:25 And so in doing so, this will allow us to get the uncertainties.

00:31 And we'll see what happens when n approaches zero, basically when n is equal to zero.

00:39 So we'll start with x, with its expectation value, and we'll see what happens basically in getting a contradiction for the uncertainty principle.

00:52 So the expectation value is obtained by putting the operator x in the integral times, in this case the square of the wave function, and we obtain the following integral, which you can obtain just by looking at a table of integrals.

01:10 It's kind of a complicated integral.

01:12 You can also do this integral with integration by parts.

01:16 But this is the integral we obtain on the interval from zero to l.

01:29 And i simplify this further.

01:32 In the next line, i'm going to go ahead and scroll down.

01:51 And we'll basically just keep going and we'll get the final result.

01:55 And we see that for the expectation value of x, it's really just just l over 2, which is what we would expect for a particle on a box of length l.

02:07 So on average, a particle is halfway along.

02:11 It's somewhere in that interval, on average, halfway.

02:15 So now we'll go on and get the expectation value for x squared.

02:21 So we just substitute in x squared inside the integral and multiply by the square of the wave function and integrate from 0 to l.

02:28 And we get again another complicated integral, but we can use a table of integrals or use integration by parts, and we get the following.

02:47 So notice in this integral that a lot of it will go away.

02:52 You have the sign multiplying this quantity in parentheses involving x squared times l and l cubed, but because of the sine function, it's going to be zero at x equal to zero as well as x equal to l.

03:08 And you'll know that because you have a 2 pi times in inside of the sign function.

03:15 So automatically those terms will vanish because they multiply by that sign function.

03:26 And again, all of that is evaluated at 0 and l.

03:30 But this greatly simplifies.

03:32 And we have the following that comes about.

03:48 And this is the final result we get...

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