From Special Relativity to Feynman Diagrams.pdf

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

Info icon This preview shows pages 525–529. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 525

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 526
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
Image of page 527

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 ;
Image of page 528
Image of page 529
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern