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ft1ln01_08

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

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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 .

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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 µ φ ∂ µ φ 2 m 2 φ 2 (2a) = 1 φ ˙ 2 1 i φ i φ 1 m 2 φ 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 δ 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 | ψ ( t ) = H | ψ ( t ) . (7) ∂t If H is independent of time, Eq. (7) has the formal solution | ψ ( t ) = e iHt/h ¯ | ψ (0) . (8) Given any operator , its expectation value in the state | ψ ( t ) is then given by ψ ( t ) | | ψ ( t ) = ψ (0) e iHt/h ¯ e iHt/h ¯ ψ (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. We will see later that the a 0 limit is problematic for interacting theories, but we will see here that this program can be carried out easily for the free theory.

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

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