Space-Time Symmetries in Quantum Physics
The transformations in space and in time which belong to the Galilei group play an important role in quantum theory. In some respect and for some aspects, their role is new as compared to classical mechanics.
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Space-Time Symmetries in Quantum Physics
Introduction The transformations in space and in time which belong to the Galilei group play an important role in quantum theory. In some respect and for some aspects, their role is new as compared to classical mechanics. Rotations, translations, and space reflection induce unitary transformations of those elements of Hilbert space which are defined with respect to the physical space R3 and to the time axis Rt . Reversal of the arrow of time induces an antiunitary transformation in H. Invariance of the Hamiltonian H of a quantum system under Galilei transformations implies certain properties of its eigenvalues and eigenfunctions which can be tested in experiment. This chapter deals, in this order, with rotations in R3 , space reflection, and time reversal. A further and more detailed analysis of the rotation group is the subject of Chap. 6 in Part Two.
4.1
The Rotation Group (Part 1)
Consider a Hilbert space and a countably infinite basis {ϕν (x)} thereof. The functions {ϕν (x)} are defined over the physical space R3 . Being elements of H they are orthogonal and normalized to 1. Given a physical wave function ψ = ν ϕν aν , the vector (a1 , a2 , . . .)T is a specific representation of this state. Every transformation R ∈ SO(3), or R ∈ O(3), interpreted as a passive transformation in R3 (i.e. a rotation of the frame of reference) induces a unitary transformation in H such that {aν } −→ aμ = Dμν (Θi )aν ; DD† = D† D = 1 . (4.1) ν
As the physical state ψ does not depend on the base used for its expansion, this implies that the base functions are contragredient to the expansion coefficients, that is to say, transform according to (D−1 )T .
F. Scheck, Quantum Physics, DOI: 10.1007/978-3-642-34563-0_4, © Springer-Verlag Berlin Heidelberg 2013
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4.1.1
4 Space-Time Symmetries in Quantum Physics
Generators of the Rotation Group
The infinite dimensional matrices D depend on the Euler angles {Θi } ≡ (φ, θ, ψ), or any other parametrization of the rotation in space. They are elements of the unitary, in general reducible, representations of the rotation group in Hilbert space. The eigenh.o. (r )Y ( x functions Rn m ˆ ) of the spherical oscillator, Sect. 1.9.4, provide an example of an orthonormal system which is defined with reference to a coordinate system in R3 . An example for the unitary transformation which is induced by a rotation about the 3-axis was discussed in Sect. 3.2.1. The elements of S O(3) are continuous functions of the angles and can be continuously deformed into the identical mapping 1. Therefore, they can be written as exponential series in three angles and three generators for infinitesimal rotations, see [Scheck (2010)]. If, for example, we choose a Cartesian basis in the physical space, denote the angles by ϕ = (ϕ1 , ϕ2 , ϕ3 ), and the generators by J = (J1 , J2 , J3 ), where the matrices Jk are given by ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 00 0 0 01 0 −1 0 J1 = ⎝ 0 0 −1 ⎠ , J2 = ⎝ 0 0 0 ⎠ , J3 = ⎝ 1 0 0 ⎠ , 01 0 −1 0 0 0 0 0 then a passive rotation in R3 read
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