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# hw33(1) - Homework Set No 3 Physics 880.08 Deadline –...

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Unformatted text preview: Homework Set No. 3, Physics 880.08 Deadline – Wednesday, May 18, 2011 1. a. (5 pts) By explicitly expanding the exponentials on the left-hand-side and carrying out the Grassmann integrals show that the following relation holds integraldisplay d ¯ χ 1 dχ 1 d ¯ χ 2 dχ 2 exp bracketleftBigg- summationdisplay i,j a ij ¯ χ i χ j bracketrightBigg exp bracketleftBigg summationdisplay k (¯ χ k ξ k + ¯ ξ k χ k ) bracketrightBigg = (det A ) exp bracketleftBigg summationdisplay i,j ¯ ξ i A- 1 ij ξ j bracketrightBigg where χ i and ξ j are Grassmann variables, and A is a 2 × 2 Hermitean matrix with elements a ij . b (5 pts) Show that- i ∂ ∂ ¯ ξ F = χ F = F χ i ∂ ∂ξ F = ¯ χ F = F ¯ χ for the function F = exp bracketleftbig i ( ¯ ξ χ + ¯ χ ξ ) bracketrightbig . Here χ and ξ are Grassmann variables. 2. Nature of the perturbation series. Consider a zero-dimensional “field theory” defined by the “path integral” I ( m, λ ) = ∞ integraldisplay-∞...
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hw33(1) - Homework Set No 3 Physics 880.08 Deadline –...

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