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Published byCiera Storer Modified over 3 years ago

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**The electromagnetic (EM) field serves as a model for particle fields**

= charge density, J = current density

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**4-vector representation of EM field**

A is the vector potential is the electrostatic potential; c = speed of light

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**How E and B are related to**

Derivation of E E+A/t = -

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x B = 0J + (1/c2) E/t x (x A) = 0J + (1/c2) /t [- - A/t] [ A] -2 A = 0J - 2A/(ct)2 - (1/c)[/(ct)] -2 A+ 2A/(ct)2 + [A+ (1/c) /(ct)] = 0J = 0 (Lorentz gauge)

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** A = - (1/c) /(ct) E = /0 [- - A/t] = /0**

-2 - /t ( A ) = /0 -2 2 /(ct)2 = /0 A = - (1/c) /(ct) (Lorentz gauge)

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**Wave equations for A and **

…. from derivations on last two slides: Lorentz gauge! A+ (1/c) /(ct) = 0

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The µ µ operator Summation implied!

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**The wave equations for A and **

can be put into 4-vector form:

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**The sources: charges and currents**

The sources represent charged particles (electrons, say) and moving charges . They describe how charges affect the EM field!

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**represents four 4-dim partial differential**

equations – in space and time!

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**If there are no electrons or moving charges:**

This is a “free” field equation – nothing but photons! We will look at solutions to these equations.

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Solutions to 1. We have seen how Maxwell’s equations can be cast into a single wave equation for the electromagnetic 4-vector, Aµ . This Aµ now represents the E and B of the EM field … and something else: the photon! If Aµ is to represent a photon – we want it to be able to represent any photon. That is, we want the most general solution to the equation:

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**First we assume the following form,**

and is a 4-vector! We are left with solving the following equation:

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Note that = - k k exp ( i k r)

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Likewise:

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**This “Fourier expansion” of the photon operator is called **

creation operator annihilation operator Spin vector Spin vector This “Fourier expansion” of the photon operator is called “second quantization”. Note that the solution to the wave equation consists of a sum over an infinite number of “photon” creation and annihilation terms. Once the ak± are interpreted as operators, the A becomes an operator.

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The A is not a “photon”. It is the photon field “operator”, which stands ready to “create” or “annihilate” a photon of any energy, , and momentum, k.

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**Recall that in order to satisfy the four wave equations the**

following had to be satisfied. This gives a relativistic four-momentum for a particle with zero rest mass! Note that = |k| c = (m0c)2 “rest” frame

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**creation & annihilation operators**

The procedure by which quantum fields are constructed from individual particles was introduced by Dirac, and is (for historical reasons) known as second quantization. Second quantization refers to expressing a field in terms of creation and annihilation operators, which act on single particle states: |0> = vacuum, no particle |p> = one particle with momentum vector p

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Notice that for the EM field, we started with the E and B fields –and showed that the relativistic “field” was a superposition of an infinite number of individual “plane wave” particles, with momentum k . The second quantization fell out naturally. This process was, in a sense, the opposite of creating a field to represent a particle. The field exists everywhere and permits “action at a distance”, without violating special relativity. We are familiar with the E&M field!

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**In this course we will only use some definitions and operations**

In this course we will only use some definitions and operations. These are one particle states. It is understood that p = k. Our definition of a one particle state is |k>. We don’t know where it is. This is consistent with a plane wave state which has (exactly) momentum p: x px /2 The following give real numbers representing the probability of finding one particle systems – or just the vacuum (no particle). vacuum one particle This is a real zero!

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**Here is a simple exercise:**

On the other hand,

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**How might we calculate a physically meaningful number from all this?**

Some things to think about: The field is everywhere – shouldn’t we integrate over all space? Then, shouldn’t we make the integration into some kind of expectation value – as in quantum mechanics? It has to be real, so maybe we should use A*A or something similar? How about <k| A*A dV |k> ? What do you think? Maybe we should start with a simpler field to work all this out! The photon field is a nice way to start, but it has spin – and a few complications we can postpone for now.

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**One final comment on the electromagnetic field: conservation of charge**

The total charge flowing out of a closed surface / sec = rate of decrease of charge inside a Lorentz invariant!

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