From Special Relativity to Feynman Diagrams.pdf

# Of 1 2 3 we see that the total contribution sum up to

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of 1, 2, 3, we see that the total contribution sum up to zero: o f 1 o q i o f 2 o q j o 2 f 3 o p i o p j ± o f 3 o q j o 2 f 2 o p i o p j ¸ ¹ þ o f 2 o q i o f 3 o q j o 2 f 1 o p i o p j ± o f 1 o q j o 2 f 3 o p i o p j ¸ ¹ þ o f 3 o q i o f 1 o q j o 2 f 2 o p i o p j ± o f 2 o q j o 2 f 1 o p i o p j ¸ ¹ ¼ 0 R. D’Auria and M. Trigiante, From Special Relativity to Feynman Diagrams , UNITEXT, DOI: 10.1007/978-88-470-1504-3, Ó Springer-Verlag Italia 2012 545

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The same of course would happen if we considered all the other terms bilinear in the first derivatives with respect to two p 0 i s and to one q i and one p i : Therefore the total sum is identically zero. 546 Appendix D: Jacobi Identity for Poisson Brackets
Appendix E Induced Representations and Little Groups E.1 Representation of the Poincaré Group The single particle states j p ; r i are constructed as a basis of a (infinite dimensional) space V ð c Þ supporting a unitary, irreducible representation of the Poincaré group. This construction is effected through the method of induced representations : we start defining the single particle states j " p ; r i in a fixed reference frame S 0 , where the four momentum is a standard one p l ¼ " p l : These states differ by the internal degree of freedom, labeled by r , related to the spin of the particle and which is acted on by the little group G ð 0 Þ & SO ð 1 ; 3 Þ of the momentum " p ³ ð " p l Þ (spin group), consisting of the Lorentz transformations K ð 0 Þ which leave " p inert: K ð 0 Þ 2 G ð 0 Þ , K ð 0 Þ l m " p m ¼ " p l : ð E : 1 Þ A transformation K ð 0 Þ of G ð 0 Þ is implemented on the states j " p ; r i by a unitary operator U ð K ð 0 Þ Þ which then maps j " p ; r i into an eigenstate of the four-momentum corresponding to the same eigenvalue " p : The vector U ð K ð 0 Þ Þj " p ; r i has then to be a linear combination of the basis elements j " p ; r i through a matrix R ³ ðR r s Þ : U ð K ð 0 Þ Þj " p ; r i ¼ Rð K ð 0 Þ Þ s r j " p ; s i : ð E : 2 Þ Such matrix K ð 0 Þ Þ defines a (unitary) representation R of G ð 0 Þ which characterizes the spin of the particle. For a massive particle m 2 0 ; G ð 0 Þ ¼ SU ð 2 Þ , see Sect. E.2 , and R has dimension 2 s þ 1 (that is r ¼ 1 ; . . . ; 2 s þ 1 Þ ; s being the spin of the particle (in units " h ); for a massless particle, m 2 ¼ 0 ; G ð 0 Þ is effectively SO ð 2 Þ , generated by the helicity operator, see Sect. E.2 , and r ¼ 1 ; 2 labels the helicity state. Proper Lorentz transformations do not alter the eigenvalue of the helicity, as proven in Sect. 9.4.2 . R. D’Auria and M. Trigiante, From Special Relativity to Feynman Diagrams , UNITEXT, DOI: 10.1007/978-88-470-1504-3, Ó Springer-Verlag Italia 2012 547

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A state j p ; r i , corresponding to a generic four momentum p ³ ð p l Þ is defined by acting on j " p ; r i with the Lorentz boost K p which relates S 0 to the RF S in which the momentum of the particle is p : p ¼ K " p : If U ð K Þ is the unitary transformation implementing a Lorentz transformation K on the states, j p ; r i is then defined as: j p ; r i ¼ j K p " p ; r i ³ U ð K p Þj " p ; r i : ð E : 3 Þ The above relation defines j p ;
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• Fall '17
• Chris Odonovan

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