Quantum Mechanics for Mathematicians
number and a particle number operatorN̂, but the lowest energy| 0 〉may not have the properties N̂| 0 〉= 0, e−iθN̂| 0 〉=| 0 〉 Ins ...
methods can be used to compute a power-series approximation in the small pa- rameter. This is an important topic in physics, cov ...
Chapter 40 Minkowski Space and the Lorentz Group For the case of non-relativistic quantum mechanics, we saw that systems with an ...
wave equation will in general be reducible. Irreducible such representations will be the objects corresponding to elementary par ...
with the last form of this using the Einstein summation convention. One motivation for introducing both upper and lower indices ...
x 0 x 1 x 2 x 0 = 0 (plane) |v|^2 < 0 (timelike) |v|^2 = 0 (light cone) |v|^2 > 0 (spacelike) Figure 40.1: Light cone stru ...
components arise by multiplication of elements inSO(3,1) byP,T,PTwhere P= 1 0 0 0 0 − 1 0 0 0 0 − 1 0 0 0 0 − 1 ...
and recall that these satisfy theso(3) commutation relations [l 1 ,l 2 ] =l 3 , [l 2 ,l 3 ] =l 1 , [l 3 ,l 1 ] =l 2 and correspo ...
40.3 The Fourier transform in Minkowski space One can define a Fourier transform with respect to the four space-time variables, ...
have the same Lie algebra asSO(3,1), and we will sometimes refer to either group as the “Lorentz group”. Recall from chapter 6 t ...
Note that both Ω and−Ω give the same linear transformation when they act by conjugation like this. One can show that all element ...
and Spin(2,2) =SL(2,R)×SL(2,R) correspond to different so-called “real forms” of a fact about complex groups that one can get by ...
Chapter 41 Representations of the Lorentz Group Having seen the importance in quantum mechanics of understanding the repre- sent ...
Turning now toSpin(3,1) =SL(2,C), one can use the same construction using homogeneous polynomials as in theSU(2) case to get irr ...
is the change of basis matrix relating the representation and its complex conjugate. This is no longer true forSL(2,C). Conjugat ...
vectors that we saw earlier. It is a representations ofSO(3,1) as well as SL(2,C). (^12 ,0)⊕(0,^12 ): This reducible 4 complex ...
Writing elements of the dual as row vectors, our example above of a par- ticular Ω acts by ( ψ^1 ψ^2 ) → ( ψ^1 ψ^2 ) ei θ 2 σ 3 ...
Restricting to theSU(2) subgroup ofSL(2,C), all these representations are unitary, and equivalent. AsSL(2,C) representations, th ...
One can easily check that these satisfy the Clifford algebra relations for gener- ators of Cliff(3,1): they anticommute with eac ...
alternative standard notation to the two-component van der Waerden notation is to use the four components ofC^4 with the action ...
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