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Quantum Field Theory, James T Wheeler

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Quantum Field Theory, James T Wheeler QUANTUM FIELD THEORY by James T. Wheeler Contents 1 From classical particles to quantum fields 3 1.1 Hamiltonian Mechanics . 3 1.1.1 Poisson brackets . 5 1.1.2 Canonical transformations . 7 1.1.3 Hamilton’s equations from the action . 9 1.1.4 Hamilton’s principal function and the Hamilton-Jacobi equation . 10 1.2 Canonical Quantization . 12 1.3 Continuum mechanics . 17 1.4 Canonical quantization of a field theory . 21 1.5 Special Relativity . 23 1.5.1 The invariant interval . 23 1.5.2 Lorentz transformations . 25 1.5.3 Lorentz invariant tensors . 29 1.5.4 Discrete Lorentz transformations . 31 1.6 Noether’s Theorem . 33 1.6.1 Example 1: Translation invariance . 35 1.6.2 Example 2: Lorentz invariance . 37 1.6.3 Symmetric stress-energy tensor (Scalar Field): . 40 1.6.4 Asymmetric stress-energy vector field . 41 2 Group theory 49 2.1 Lie algebras . 55 2.2 Spinors and the Dirac equation . 65 2.2.1 Spinors for O(3) .............................. 65 2.2.2 Spinors for the Lorentz group . 68 2.2.3 Dirac spinors and the Dirac equation . 71 2.2.4 Some further properties of the gamma matrices . 79 2.2.5 Casimir Operators . 81 3 Quantization of scalar fields 84 3.1 Functional differentiation . 84 3.1.1 Field equations as functional derivatives . 88 3.2 Quantization of the Klein-Gordon (scalar) field . 89 3.2.1 Solution for the free quantized Klein-Gordon field . 92 1 3.2.2 Calculation of the Hamiltonian operator . 96 3.2.3 Our first infinity . 99 3.2.4 States of the Klein-Gordon field . 99 3.2.5 Poincaré transformations of Klein-Gordon fields . 101 3.3 Quantization of the complex scalar field . 103 3.3.1 Solution for the free quantized complex scalar field . 105 3.4 Scalar multiplets . 109 3.5 Antiparticles . 110 3.6 Chronicity, time reversal and the Schrödinger equation . 118 4 Quantization of the Dirac field 120 4.1 Hamiltonian formulation . 120 4.2 Solution to the free classical Dirac equation . 122 4.3 The spin of spinors . 126 4.4 Quantization of the Dirac field . 130 4.4.1 Anticommutation . 134 4.4.2 The Dirac Hamiltonian . 136 4.5 Symmetries of the Dirac field . 139 4.5.1 Translations . 139 5 Gauging the Dirac action 140 5.1 The covariant derivative . 141 5.2 Gauging . 144 6 Quantizing the Maxwell field 146 6.1 Hamiltonian formulation of the Maxwell equations . 146 6.2 Handling the constraint . 150 6.3 Vacuum solution to classical E&M . 154 6.4 Quantization . 155 7 Appendices 157 7.1 Appendix A: The Casimir operators of the Poincaré group. 157 7.2 Appendix B: Completeness relation for Dirac solutions . 160 8 Changes since last time: 162 2 1 From classical particles to quantum fields First, let’s review the use of the action in classical mechanics. I’ll reproduce here a conden- sation of my notes from classical mechanics. If you’d like a copy of the full version, just ask; what’s here is more than enough for our purposes. 1.1 Hamiltonian Mechanics Perhaps the most beautiful formulation of classical mechanics, and the one which ties most closely to quantum mechanics, is the canonical formulation. In this approach, the position and velocity velocity variables of Lagrangian mechanics are replaced by the position and @L conjugate momentum, pi ≡ . It turns out that by doing this the coordinates and momenta @q_i are put on an equal footing, giving the equations of motion a much larger symmetry. To make the change of variables, we use a Legendre transformation. This may be familiar from thermodynamics, where the internal energy, Gibb’s energy, free energy and enthalpy are related to one another by making different choices of the independent variables. Thus, for example, if we begin with dU = T dS − P dV (1) where T and P are regarded as functions of S and V; we can set H = U + VP (2) and compute dH = dU + P dV + V dP (3) = T dS − P dV + P dV + V dP (4) = T dS + V dP (5) to achieve a formulation in which T and V are treated as functions of S and P: The same technique works here. We have the Lagrangian, L(qi; q_i) and wish to find a function H(qi; pi): The differential of L is N N X @L X @L dL = dq + dq_ (6) @q i @q_ i i=1 i i=1 i N N X X = p_idqi + pidq_i (7) i=1 i=1 where the second line follows by using the equations of motion and the definition of the conjugate momentum. Therefore, set N X H(qi; pi) = piq_i − L (8) i=1 3 so that N N X X dH = dpiq_i + pidq_i − dL (9) i=1 i=1 N N N N X X X X = dpiq_i + pidq_i − p_idqi − pidq_i (10) i=1 i=1 i=1 i=1 N N X X = dpiq_i − p_idqi (11) i=1 i=1 Notice that, as it happens, H is of the same form as the energy. Clearly, H is a function of the momenta. To see that we have really eliminated the dependence on velocity we may compute directly, N ! @H @ X = p q_ − L(q ; q_ ) (12) @q_ @q_ i i i i j j i=1 N X @L = p δ − (13) i ij @q_ i=1 j @L = pj − (14) @q_j = 0 (15) The equations of motion are already built into the expression above for dH: Since the differ- ential of H may always be written as N N X @H X @H dH = dq + dp (16) @q i @p i i=1 j i=1 j we can simply equate the two expressions: N N N N X X X @H X @H dH = dp q_ − p_ dq = dq + dp (17) i i i i @q i @p i i=1 i=1 i=1 i i=1 i Then, since the differentials dqi and dpi are all independent, we can equate their coefficients, @H p_i = − (18) @qi @H q_i = (19) @pj These are Hamilton’s equations. 4 1.1.1 Poisson brackets Suppose we are interested in the time evolution of some function of the coordinates, momenta and time, f(qi; pi; t): It could be any function – the area of the orbit of a particle, the period of an oscillating system, or one of the coordinates. The total time derivative of f is df X @f dqi @f dpi @f = + + (20) dt @qi dt @pi dt @t Using Hamilton’s equations we may write this as df X @f @H @f @H @f = − + (21) dt @qi @pi @pi @qi @t Define the Poisson bracket of H and f to be N X @f @H @f @H fH; fg = − (22) @q @p @p @q i=1 i i i i Then the total time derivative is given by df @f = fH; fg + (23) dt @t @f If f has no explicit time dependence, so that @t = 0; then the time derivative is given completely by the Poisson bracket: df = fH; fg (24) dt We generalize the Poisson bracket to two arbitrary functions, N X @g @f @g @f ff; gg = − (25) @q @p @p @q i=1 i i i i The importance of the Poisson bracket stems from the underlying invariance of Hamil- tonian dynamics. Just as Newton’s second law holds in any inertial frame, there is a class of canonical coordinates which preserve the form of Hamilton’s equations. One central re- sult of Hamiltonian dynamics is that any transformation that preserves certain fundamental Poisson brackets is canonical, and that such transformations preserve all Poisson brackets Since the properties we regard as physical cannot depend on our choice of coordinates, this means that essentially all truly physical properties of a system can be expressed in terms of Poisson brackets. In particular, we can write the equations of motion as Poisson bracket relations. Using the general relation above we have dq i = fH; q g (26) dt i 5 N X @qi @H @qi @H = − (27) @q @p @p @q j=1 j j j j N X @H = δ (28) ij @p j=1 j @H = (29) @pi and dp i = fH; p g (30) dt i N X @pi @H @pi @H = − (31) @q @p @p @q j=1 j j j j @H = − (32) @qi @qi Notice that since qi; pi and are all independent, and do not depend explicitly on time, = @pj @pi = 0 = @qi = @pi : @qj @t @t We list some properties of Poisson brackets. Bracketing with a constant always gives zero ff; cg = 0 (33) The Poisson bracket is linear faf1 + bf2; gg = aff1; gg + bff2; gg (34) and Leibnitz ff1f2; gg = f2ff1; gg + f1ff2; gg (35) These three properties are the defining properties of a derivation; which is the formal gen- eralization of differentiation. The action of the Poisson bracket with any given function f on the class of all functions, ff; ·} is therefore a derivation. If we take the time derivative of a bracket, we can easily show @ @f @g ff; gg = f ; gg + ff; g (36) @t @t @t The bracket is antisymmetric ff; gg = − fg; fg (37) and satisfies the Jacobi identity, ff; fg; hgg + fg; fh; fgg + fh; ff; ggg = 0 (38) 6 for all functions f; g and h: These properties are two of the three defining properties of a Lie algebra (the third defining property of a Lie algebra is that the set of objects considered, in this case the space of functions, be a finite dimensional vector space, while the space of functions is infinite dimensional).
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