582-2008-7-final

# 582-2008-7-final - Physics 582 Fall Semester 2008 Professor...

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Unformatted text preview: Physics 582, Fall Semester 2008 Professor Eduardo Fradkin Problem Set No. 7/ Final Exam Due Date: Friday, December 19, 2008; 5:00 pm This is a take home final exam. As noted in the course website you will have to send me your solutions by email. This can be done either by writing your solutions in LaTeX and sending me the pdf file, or by scanning your handwritten solutions and sending me them as a pdf file. If you choose the latter option you will have to make sure that the scanned file is clearly legible and this will require the use of a dark pen and clear handwriting. Your solutions will have to be sent to me (not to the TA) by email before Friday December 19, 5:00 pm CST, without exception. No solutions will be accepted after that. In this set you will study the properties of a theory of a self-interacting complex scalar field φ ( x ), in perturbation theory. The Lagrangian density for this theory in four-dimensional Minkowski space-time is L = 1 2 ∂ μ φ ∗ ( x ) ∂ μ φ ( x ) − 1 2 m 2 φ ∗ ( x ) φ ( x ) − λ 4! ( φ ∗ ( x ) φ ( x )) 2 + J ( x ) ∗ φ ( x )+ J ( x ) φ ∗ ( x ) (1) Assume that m 2 > 0 and that coupling constant λ > 0. Here J ( x ) are a set of (complex) sources. Here you will be asked to do calculations using both canonical (operator) and path integral methods. 1. Derive the classical equations of motion of this theory. Give an argument to show that the classical ground state of the theory is given by the classical field configuration ¯ φ ( x ) = 0. 2. Use the canonical (operator) formalism to quantize this theory. Define the canonical momentum which is conjugate to the field φ ( x ) and write down the equal time commutation relations satisfied by the canonical pairs. Derive the quantum mechanical Hamiltonian for this theory and write down an explicit expression for the field expansion for the free field case....
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582-2008-7-final - Physics 582 Fall Semester 2008 Professor...

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