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

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Quantum electrodynamics I. MAXWELL’S 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 = μ 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 0 E 2 + B 2 μ 0 (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 Maxwell’s 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 Maxwell’s equations are solved provided E = - A ∂t , (4)
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2 and A obeys the equations ∇ · A = 0 2 A = 1 c 2 2 A ∂t 2 . (5) Here c = 1 μ 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 oscillators—the ‘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 π 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 π ) 3 (8) which holds in the limit V → ∞ .
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3 The Hamiltonian of these oscillators is H = 1 2 X k s ( p 2 k s + ω 2 k q 2 k s ) , (9) where the frequency ω 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 = - ω 2 k q k s . (11) We can now make our main claim. The Hamiltonian (9) and the equations of motion (11) are exactly equivalent to the Hamiltonian (2) and Maxwell’s equations (1).
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