Quantum Mechanics for Mathematicians
A third representation, the Majorana representation, is given by (now no longer writing in 2 by 2 block form, but as 4 by 4 matr ...
Chapter 42 The Poincar ́e Group and its Representations In chapter 19 we saw that the Euclidean groupE(3) has infinite dimension ...
onn-component wavefunctions, solutions to a wave equation (hereSis anndimensional representation of the Lorentz group). These wi ...
[k 1 ,k 2 ] =−l 3 , [k 3 ,k 1 ] =−l 2 , [k 2 ,k 3 ] =−l 1 or [lj,lk] =jklll, [kj,kk] =−jklll and that the commutation relation ...
Note that infinitesimal boosts do not commute with infinitesimal time trans- lation, so after quantization boosts will not commu ...
To construct representations ofP, we would like to generalize this construc- tion fromR^3 to Minkowski spaceM^4. To do this, one ...
A straightforward calculation using the Poincar ́e Lie algebra commutation relations shows thatP^2 is a Casimir operator since o ...
Given an irreducible representation, the operatorP^2 will act by the scalar −p^20 +p^21 +p^22 +p^23 which can be positive, negat ...
p 0 m p 3 m −m −m O 0 O(m, 0 , 0 ,0) O(1, 0 , 0 ,1) O(− 1 , 0 , 0 ,1) O(0, 0 , 0 ,m) O(−m, 0 , 0 ,0) Figure 42.1: Orbits of vect ...
whereR∈SO(3), soK(m, 0 , 0 ,0)=SO(3). Irreducible representations of this group are classified by the spin. For spin 0, points o ...
satisfying the equation −p^20 +p^21 +p^22 +p^23 =m^2 It is not too difficult to see that the stabilizer group of the orbit isK(0 ...
and calculating the commutators [b 1 ,b 2 ] = 0, [l 3 ,b 1 ] =b 2 , [l 3 ,b 2 ] =−b 1 we see that these three elements of the Li ...
42.4 For further reading For an extensive discussion of the Poincar ́e group, its Lie algebra and repre- sentations, see [97]. W ...
Chapter 43 The Klein-Gordon Equation and Scalar Quantum Fields In the non-relativistic case we found that it was possible to bui ...
that such irreducible representations are characterized in part by the scalar value that the Casimir operatorP^2 takes on the re ...
Solutions to this will be distributions that are non-zero only on the hyperboloid p^20 −p^21 −p^22 −p^23 −m^2 = 0 in energy-mome ...
Besides specifying the functionψ(x,t), elements of the single-particle spaceH 1 could be uniquely characterized in two other way ...
with the left-hand side an explicitly Lorentz invariant measure (sincep^20 −ωp^2 = −p^2 −m^2 is Lorentz invariant). An arbitrary ...
Φ(φ(x),π(x)): the solution with initial dataφ(x),π(x) att= 0. A(α(p)): the solution with initial data att= 0 specified by the c ...
however much more convenient, as in the non-relativistic case of chapters 36 and 37, to use the Bargmann-Fock representation, tr ...
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