QED - Quantum electrodynamics I. MAXWELLS EQUATIONS We...

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Unformatted text preview: Quantum electrodynamics I. MAXWELLS EQUATIONS We begin with the equations for the electric field E ( r ,t ) and magnetic field B ( r ,t ) in the vacuum: E =- B t B = 0 0 E t E = 0 B = 0 (1) We will also need the expression for the energy of the electromagnetic field H = 1 2 Z d 3 r E 2 + B 2 (2) Note that in the above equations r is merely the label of a point in space at which the fields are measured; it is not the co-ordinate of anything. It is the E and B fields which evolve in time, and so these will eventually become the co-ordinates and momenta of our theory. It is useful to partially solve Maxwells equations by introducing a vector potential A so that B = A ; (3) this can be view as the most general solution to B = 0. Then it is easy to check that the remaining Maxwells equations are solved provided E =- A t , (4) 2 and A obeys the equations A = 0 2 A = 1 c 2 2 A t 2 . (5) Here c = 1 0 0 (6) is the velocity of light. II. CLASSICAL NORMAL MODE ANALYSIS We will now rewrite Eqs (1), (2), (5) using a new set of variables. Note that we will not introduce quantum mechanics here. This section does not introduce any physics that is not already in Section I; it only writes things in a different way. To make the degrees of freedom countable (but still infinite), we place the fields in a L L L box with periodic boundary conditions. In the end, we will send the volume V = L 3 to infinity. In this box, our new degrees of freedom are the co-ordinates q k s and momenta p k s of an infinite set of independent harmonic oscillators. You should think of these as abstract oscillatorsthe co-ordinate q k s is not connected to the position in space of anything. These oscillators are labelled by the wavevector k and the polarization s = 1 , 2. Again note that k is merely a label; because of the periodic boundary conditions, it is restricted to the values k = 2 L ( n 1 ,n 2 ,n 3 ) , n 1 , 2 , 3 = integers (7) The only feature of these values we use is the mapping X k V Z d 3 k (2 ) 3 (8) which holds in the limit V . 3 The Hamiltonian of these oscillators is H = 1 2 X k s ( p 2 k s + 2 k q 2 k s ) , (9) where the frequency k = c | k | . (10) The equations of motion of the oscillators are the familiar ones associated with simple harmonic motion dq k s dt = p k s dp k s dt =- 2 k q k s . (11) We can now make our main claim. The Hamiltonian (9) and the equations of motionWe can now make our main claim....
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This note was uploaded on 02/15/2011 for the course PHYS 143B taught by Professor Unknow during the Spring '10 term at Harvard.

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QED - Quantum electrodynamics I. MAXWELLS EQUATIONS We...

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