## PHY450H1S. Relativistic Electrodynamics Lecture 9 (Taught by Prof. Erich Poppitz). Dynamics in a vector field.

Posted by peeterjoot on February 3, 2011

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# Reading.

Covering chapter 2 material from the text [1].

Covering lecture notes pp. 56.1-72: comments on mass, energy, momentum, and massless particles (56.1-58); particles in external fields: Lorentz scalar field (59-62); reminder of a vector field under spatial rotations (63) and a Lorentz vector field (64-65) [Tuesday, Feb. 1]; the action for a relativistic particle in an external 4-vector field (65-66); the equation of motion of a relativistic particle in an external electromagnetic (4-vector) field (67,68,73) [Wednesday, Feb. 2]; mathematical interlude: (69-72): on 3×3 antisymmetric matrices, 3-vectors, and totally antisymmetric 3-index tensor – please read by yourselves, preferably by Wed., Feb. 2 class! (this is important, well also soon need the 4-dimensional generalization)

# More on the action.

Action for a relativistic particle in an external 4-scalar field

Unfortunately we have no 4-vector scalar fields (at least for particles that are long lived and stable).

PICTURE: 3-vector field, some arrows in various directions.

PICTURE: A vector in an frame, and a rotated (counterclockwise by angle ) frame with the components in each shown pictorially.

We have

More generally we have

Here is an matrix rotating

A four vector field is , with and we’d write

Now is an matrix. Our four vector field is then

We have

From electrodynamics we know that we have a scalar field, the electrostatic potential, and a vector field

What’s a plausible action?

How about

This isn’t translation invariant.

Next simplest is

Could also do

but it turns out that this isn’t gauge invariant (to be defined and discussed in detail).

Note that the convention for this course is to write

Where is dimensionless (). Some authors use

The simplest action for a four vector field is then

(Recall that ).

In this action is nothing but a Lorentz scalar, a property of the particle that describes how it “couples” (or “feels”) the electrodynamics field.

Similarily is a Lorentz scalar which is a property of the particle (inertia).

It turns out that all the electric charges in nature are quantized, and there are some deep reasons (in magnetic monopoles exist) for this.

Another reason for charge quantitization apparently has to do with gauge invariance and associated compact groups. Poppitz is amusing himself a bit here, hinting at some stuff that we can eventually learn.

Returning to our discussion, we have

with the electrodynamics four vector potential

Here and are summed over.

For the other half of the Euler-Lagrange equations we have

Equating these, and switching to coordinates for 2.23, we have

A final rearrangement yields

We can identity the second term with the magnetic field but first have to introduce antisymmetric matrices.

# antisymmetric matrixes

where

latex \mu\nu\lambda$} \\ -1 & \quad \mbox{for odd permutations of }\end{array}\end{aligned} \hspace{\stretch{1}}(3.27)$

Example:

We can show that

Using

we can verify the identity 3.28 by expanding

Returning to the action evaluation we have

but

So

or

\paragraph{What is the energy component of the Lorentz force equation}

I asked this, not because I don’t know (I could answer this myself from , in the geometric algebra formalism, but I was curious if he had a way of determining this from what we’ve derived so far (intuitively I’d expect this to be possible). Answer was:

Observe that this is almost a relativisitic equation, but we aren’t going to get to the full equation yet. The energy component can be obtained from

Since the full equation is

“take with a grain of salt, may be off by sign, or factors of ”.

Also curious is that he claimed the energy component of this equation was not very important. Why would that be?

# Gauge transformations.

Claim

changes by boundary terms only under

“gauge transformation” :

where is a Lorentz scalar. This is the four gradient. Let’s see this

Therefore the equations of motion are the same in an external and .

Recall that the and fields do not change under such transformations. Let’s see how the action transforms

Observe that this last bit is just a chain rule expansion

so we have

This allows the line integral to be evaluated, and we find that it only depends on the end points of the interval

which completes the proof of the claim that this gauge transformation results in an action difference that only depends on the end points of the interval.

\paragraph{What is the significance to this claim?}

# References

[1] L.D. Landau and E.M. Lifshits. *The classical theory of fields*. Butterworth-Heinemann, 1980.

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