Quantum Mechanics for Mathematicians
the operatorsP̂andL̂, an integral of a product involving one creation and one annihilation operator. The boost operators will be ...
Chapter 45 U(1) Gauge Symmetry and Electromagnetic Fields We have now constructed both relativistic and non-relativistic quantum ...
coordinatesx. Such phase transformations are called “gauge transformations” and form an infinite dimensional group under pointwi ...
The vector potential allows one to define a new kind of derivative, such that the derivative of the fieldψhas the same homogeneo ...
Note that whileDAμis a differential operator, [DAμ,DAν] and thus the curvature is just a multiplication operator. The electromag ...
Another exactly solvable special case is that of a constant magnetic field B= (0, 0 ,B), for which one possible choice of vector ...
The first two equations are for non-relativistic theories, and one can interpret these equations as describing a single quantum ...
Stokes theorem for differential forms implies that ifγis a closed curve, and γis the boundary of a surfaceS(γ=∂S), then ∫ γ A= ∫ ...
γ(0) γ(τ) γ(1) C ψ(γ(1)) eie ∫ γAψ(γ(0)) e ∫ γA ψ(γ(0)) Figure 45.1: Comparing a complex field at two points in a gauge invarian ...
definition of the curvature, where new terms with different behavior arise, due to the non-commutative nature of the group. The ...
and magnetic field strength components as in theU(1) case, the equations for matrix-valued electric and magnetic fields are: Ej( ...
Chapter 46 Quantization of the Electromagnetic Field: the Photon Understanding the classical field theory of coupled dynamical s ...
In this chapter we will see how to make theAμfields dynamical variables, although restricting to the special case of free electr ...
46.2 The Hamiltonian formalism for electromag- netic fields In order to quantize the electromagnetic field, we first need to exp ...
Requiring that these coordinates behave just like position and momentum co- ordinates in the finite dimensional case, we can spe ...
then equation 46.11 can be written ∂E ∂t (t,x) =∇A×B The problem with these equations is that they are non-linear equations inA, ...
if we defineHphysas the subset of states satisfying 46.12, this can be identified with the space of wavefunctions of two positio ...
and an action of the groupG 0 of time-independent gauge transformations, with Lie algebra the functionsφ(x). In this case the co ...
the Coulomb gauge condition automatically implies that Gauss’s law (46.4) will hold. We thus can take as phase space the solutio ...
They provide an orthonormal basis of the tangent space atpto the sphere of radius|p|. p 2 (p) 1 (p) Figure 46.1: Polarizatio ...
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