From Special Relativity to Feynman Diagrams.pdf

Labels internal degrees of freedom which can be made

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labels internal degrees of freedom which can be made to freely vary by means of Poincaré transformations at fixed point x μ in M 4 . We may convince ourselves that the largest group of transformations which leaves x μ fixed is the full Lorentz group. 12 Consider the state corresponding to the origin of the RF x μ 0 and act on it by means of a Lorentz transformation: | 0 , α −→ U ( ) | 0 , α D β α | 0 , β . (9.96) In virtue of homogeneity of space–time, whatever statement about the point x μ = 0 equally applies to any other point x μ . We conclude that the matrix D ( D β α ) = 12 This statement seems to be at odds with what we have learned from our earlier discussion about Lorentz transformations: Under a Lorentz transformation a generic position four-vector x μ transforms into a different one x μ = μ ν x ν , and the only four vector which is left invariant is the null one ( x μ ) 0 = ( 0 , 0 , 0 , 0 ) defining the origin of the RF. For a given Lorentz transformation in SO ( 1 , 3 ) and a point P described by x ( x μ ), we can define the Poincaré transformation x ( 1 , x )( , 0 )( 1 , x ), see Sect.4.7.2 of Chap.4 for the notation, which consists in a first translation ( 1 , x ) mapping the origin O into O = P ( x 0 ), then a Lorentz transformation which leaves O invariant 0 0 , followed by a second translation which brings back O into O ( 0 x ). By construction x , which is not pure Lorentz since it contains translations, leaves x invariant. The transformations x , corresponding to SO ( 1 , 3 ), close a group which has the same structure as the Lorentz group , though being implemented by different transformations: The correspondence x for a given x is one-to-one, and, moreover, if = then x x = x . The two groups are said to be isomorphic . Transformation groups sharing the same structure represent the same symmetry. We shall denote the group consisting of the x transformation by SO ( 1 , 3 ) x . It can be regarded as the copy of the Lorentz group, depending on the point x , which leaves x invariant.
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290 9 Quantum Mechanics Formalism D ( ) acting on the internal index α is a representation of the Lorentz group . Assum- ing {| x , α } to be a basis of the Hilbert space we define the coordinate representation of a state by expanding it in this basis: | a = d 4 x α α ( x ) | x , α . (9.97) Acting on | a by means of U ( , x 0 ) we deduce the transformation property of the coefficients α ( x ) : | a = U ( , x 0 ) | a = d 4 x α ( x ) U ( , x 0 ) | x , α = d 4 x α ( x ) D β α | x x 0 , β = d 4 x α ( x ) D β α | x , β = d 4 x β ( x ) | x , β , (9.98) where x x x 0 and we have used the invariance of the elementary space-time volume under Poincaré transformations: d 4 x = d 4 x . We conclude that α ( x ) = D α β β ( x ). (9.99) We have thus retrieved the general transformation property ( 7.47 ) of a relativistic field under Poincaré transformations. As we did in Sect.7.4.2 , we can describe the effect of a Poincaré transformation ( , x 0 ) on α ( x ) in terms of the active action of an operator O ( , x 0 ) , as in ( 7.90 ): α ( x ) ( , x 0 ) −→ α ( x ) = O ( , x 0 ) α ( x ). (9.100) We then write O ( , x 0 ) as the exponential of infinitesimal differential operators, as in ( 7.91 ): O ( , x 0 ) = e i x μ 0 ˆ P μ e i 2 θ ρσ ˆ J ρσ , D ( ) α β = e i 2 θ ρσ
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