aqft_final_07 - MATHEMATICAL TRIPOS Tuesday 5 June 2007...

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Unformatted text preview: MATHEMATICAL TRIPOS Part III Tuesday 5 June 2007 9.00 to 12.00 PAPER 51 ADVANCED QUANTUM FIELD THEORY Attempt THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Cover sheet None Treasury Tag Script paper You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 1 Explain how for anti-commuting variables θ i , θ i θ j =- θ j θ i , integration may be defined so that Z d θ n . . . d θ 1 θ i 1 . . . θ i n = i 1 ...i n , 12 ...n = 1 . Show that for an n × n matrix B and with θ = ( θ 1 , . . . θ n ), ¯ θ = ( ¯ θ 1 , . . . , ¯ θ n ) anti-commuting n-vectors Z n Y i =1 d ¯ θ i d θ i exp (- ¯ θ · B θ ) = det B . Let O ( θ , ¯ θ ) = 1 det B Z n Y i =1 d ¯ θ i d θ i O ( θ , ¯ θ ) exp (- ¯ θ · B θ ) , for any polynomial O ( θ , ¯ θ ). Show that θ i ¯ θ j = ( B- 1 ) ij . What is θ i ¯ θ j θ k ¯ θ l ? Show that for n × n matrices M , N ( ¯ θ · M θ ) ( ¯ θ · N θ ) =- tr ( M B- 1 N B- 1 ) + tr ( M B- 1 ) tr ( N B- 1 ) ....
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