ft1ln01_08

ft1ln01_08 - MIT OpenCourseWare http://ocw.mit.edu 8.323...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 8.323 Relativistic Quantum Field Theory I Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.323: Relativistic Quantum Field Theory I Prof. Alan Guth February 16, 2008 LECTURE NOTES 1 QUANTIZATION OF THE FREE SCALAR FIELD As we have already seen, a free scalar field can be described by the Lagrangian L = d 3 x , (1) where 1 1 = 2 2 m 2 2 (2a) = 1 2 1 i i 1 m 2 2 . (2b) 2 2 2 Our goal is to quantize this theory, in the sense of developing a quantum theory that corresponds to the classical theory described by the above Lagrangian. 1. CANONICAL QUANTIZATION: Here we will use the method of canonical quantization, which I assume is already familiar to you in the context of quantum mechanics. Specifically, suppose that we were given a Lagrangian with a discrete number of dynamical variables q i : L = L ( q i , q i , t ) . (3) The canonical momenta would then be defined by L p i , (4) q i and the Hamiltonian would be given by H = p i q i L . (5) i A quantum theory corresponding to this classical theory could then be constructed by promoting each q i and p i to an operator on a Hilbert space, and insisting on the canonical commutation relations [ q i , p j ] = i h ij . (6) 8.323 LECTURE NOTES 1, SPRING 2008: Quantization of the Free Scalar Field p. 2 For most of this course we will use units for which h 1, but for now I will leave the h s in the equations. The Hamiltonian H ( p i , q i ) is then also an operator on the Hilbert space, and in the Schr odinger picture the physical states evolve according to the Schr odinger equation i h | ( t ) = H | ( t ) . (7) t If H is independent of time, Eq. (7) has the formal solution | ( t ) = e iHt/h | (0) . (8) Given any operator , its expectation value in the state | ( t ) is then given by ( t ) | | ( t ) = (0) e iHt/h e iHt/h (0) . (9) This equation leads naturally to the Heisenberg picture description, in which the states are treated as time-independent, and all of the time dependence is incorporated into the evolution of the operators: ( t ) = e iHt/h e iHt/h . (10) 2. FIELD QUANTIZATION BY LATTICE APPROXIMATION: To quantize the classical field theory of Eq. (2), we can begin by quantizing a lattice version of the theory. That is, we can replace the continuous space by a cubic lattice of closely spaced grid points, with a lattice spacing a , and we can truncate the space to a finite region. The system then reduces to one with a discrete number of dynamical variables, exactly like the systems that we already know how to quantize. Then if we can take the limit as the lattice spacing a approaches zero and the volume approaches infinity, the quantization of the field theory can be completed. Wecompleted....
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This note was uploaded on 11/08/2011 for the course PHY 8.323 taught by Professor Staff during the Spring '08 term at MIT.

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ft1ln01_08 - MIT OpenCourseWare http://ocw.mit.edu 8.323...

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