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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 timeindependent, 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.
 Spring '08
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 Mass, Quantum Field Theory

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