From Relativistic Quantum Mechanics and Field Theory (Franz Gross) Solve. 5-5
Posted: Wed May 18, 2022 5:02 pm
From Relativistic Quantum Mechanics and Field Theory (Franz
Gross) Solve. 5-5
It is convenient to choose the 16 basis matrices, li, so that they have well- defined transformation properties under LT's. Since the y's have such properties, we are led to choose the following 16 matrices for this basis: # matrices 1 scalar 1 ун vector 4 · 174, ") = OHV antisymmetric tensor 6 (5.129) 4 you iz Oryty2y3 = 7 = axial vector pseudoscalar 1 16 It can be seen by inspection that all of these matrices are linearly independent.
Then, for any LT A which transforms the coordinates and four-vector potential from an unprimed frame to a primed frame, r' = A. Α' (α') = ΛΑ(α) , (5.100) we seek a representation, S(A), which transforms the Dirac wave function from the unprimed to the primed coordinate system V' (I') = S(A)*(I) (5.101) Covariance is the requirement that this transformation leave the Dirac equation (5.12) invariant in form, so that in the primed frame, ([p - eA (2) - m) + (z) = 0 . This requirement determines S(A). To find the equation which defines S(A), substitute (5.101) into the above equation and multiply by S-'(A). Recall that p'* = Amvp" implies that pu = (^-) - Pu, and obtain S-4(1){zu (1-1)" - (pv – eAy(t)) – m} S(A)v(x) = 0 . : This equation is invariant in form if (4-?) S-'(A)"S(A) = que , which implies s-(A)745(A) = AMU. (5.102) = This equation will tell us how to construct the S(A).
5.5 Consider the following Dirac matrix element: a M"(x) = V(I) OH M= (2), ar where all was defined in Eq. (5.129). (a) From the structure of M, guess how it transforms under LT's. Write down the transformation law explicitly, using the notation z' = Ar. (b) Using the Lorentz transformation properties of the Dirac wave functions, Eq. (5.101), and the property Eq. (5.102), prove that your transformation law is correct or find the correct one.
Gross) Solve. 5-5
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Then, for any LT A which transforms the coordinates and four-vector potential from an unprimed frame to a primed frame, r' = A. Α' (α') = ΛΑ(α) , (5.100) we seek a representation, S(A), which transforms the Dirac wave function from the unprimed to the primed coordinate system V' (I') = S(A)*(I) (5.101) Covariance is the requirement that this transformation leave the Dirac equation (5.12) invariant in form, so that in the primed frame, ([p - eA (2) - m) + (z) = 0 . This requirement determines S(A). To find the equation which defines S(A), substitute (5.101) into the above equation and multiply by S-'(A). Recall that p'* = Amvp" implies that pu = (^-) - Pu, and obtain S-4(1){zu (1-1)" - (pv – eAy(t)) – m} S(A)v(x) = 0 . : This equation is invariant in form if (4-?) S-'(A)"S(A) = que , which implies s-(A)745(A) = AMU. (5.102) = This equation will tell us how to construct the S(A).
5.5 Consider the following Dirac matrix element: a M"(x) = V(I) OH M= (2), ar where all was defined in Eq. (5.129). (a) From the structure of M, guess how it transforms under LT's. Write down the transformation law explicitly, using the notation z' = Ar. (b) Using the Lorentz transformation properties of the Dirac wave functions, Eq. (5.101), and the property Eq. (5.102), prove that your transformation law is correct or find the correct one.