Weinberg and Ray Transformations – Ch2 App A.

[rocdoc]

I wondered if anyone might have something to say about how a 'Ray Transformation' should be defined and also point out what Weinberg means by his 'ray transformations T', see Weinberg, Reference 1, pg 91. Should a ray transformation be an equivalence class of operators?

I have started to try to read Appendix A of Chapter 2 of reference 1, are Weinbergs ray transformations , symbolised by T, the same as Bargmann’s , 'Operator Rays'?

Bargmann says ,see pg5 of reference 2

'Operator rays. A unitary operator ray $\mathbf U$ is the set of all operators (on h) of the form rUo, where Uo is a fixed unitary operator and r any complex number of modulus 1. (Any operator U contained in $\mathbf U$ will be called a representative of the ray $\mathbf U$.) In the sequel only unitary operator rays will be considered.'
He also says,see Pg6 of reference 2

$$\text{The vector ray }\mathbf{Uf} \text{ consists of all vectors }Uf (U \in \mathbf U, f \in \mathbf f).$$

So , in the above $\mathbf f$ is a vector ray, i.e. a ray of vectors which represents a physical state in a Hilbert space, h.

References

1) Steven Weinberg, The Quantum Theory of Fields, Volume I Foundations, Cambridge University Press, 1995.

2)V. Bargmann, Annals of Mathematics,Second Series,Vol59,No1,p1-46 (1954). This may be accessed via JSTOR.

 

[vanhees71]

This is an utmost important issue, glossed over in most quantum-theory textbooks. The more one must appreciate Weinberg's exceptionally good textbooks on relativistic QFT.

Contrary to many sloppy ideas stated in some textbooks, a pure state is NOT represented by a vector in Hilbert space but either by the projection operator $\hat{\rho}=|\psi \rangle \langle \psi|$ with a normalized vector $|\psi \rangle$ as the statistiscal operator or equivalently by a unit ray in Hilbert space, i.e., a normalized Hilbert-space vector $|\psi \rangle$ with all vectors of the form $\exp(\mathrm{i} \varphi) |\psi \rangle$, $\varphi \in \mathbb{R}$ identified.

By definition a symmetry is a transformation of states (represented statistical operators or rays for pure states) and observables such that the physical situation is essentially unchanged. E.g., translation invariance in space, which is a fundamental symmetry of special-relativistic and Newtonian space time, means that the natural laws are the same at any place, i.e., an experiment done at one place has the very same outcome when done in an identical way at any other place.

Mathematically symmetries are described by corresponding groups of transformations, and the very fact that a pure state is represented by a ray rather then a state vector itself implies that symmetry transformations are represented by socalled ray representations, i.e., if $g_1$ and $g_2$ are two elements of the symmetry group, then each of the corresponding transformations is represented by a unitary (or antiunitary) operator $U(g_j)$, and these operators build necessarily a unitary ray representation, i.e., there exist phases $\phi(g_1,g_2) \in \mathbb{R}$ such that
$$U(g_1 g_2)=U(g_1) U(g_2) \exp[\mathrm{i} \phi(g_1,g_2)].$$
In may cases, such ray representations can be "lifted" to unitary representations, i.e., you can redefine the phase factors of the $U(g_j)$ such that for all $g_1$ and $g_2$ the phase $\phi(g_1,g_2)=0$. This is investigated in detail by Weinberg, and it's true for the proper orthochronous Poincare group, but it's not the case for the Galileo group in non-relativistic quantum mechanics, because the true unitary representations of the Galileo group do not lead to physically interpretible quantum theoretical description of any known object in nature. The classical Galileo group can be extended to a quantum Galileo group with the mass as a socalled central charge of the corresponding Galileo Lie algebra. In contradistinction mass in the case of the proper orthochronous Poincare group mass (squared) is a Casimir operator, and this explains the subtle differences between the concept of mass in Newtonian and special-relativistic physics.

Another point which makes the ray idea so important is the fact that not the classical symmetry groups govern the dynamics of quantum systems but their universal covering groups. For the Galileo group as well s the proper orthochronous Poincare group this leads to the substitution of the classical rotation subgroup SO(3), which represents the isotropy of space in both cases, by its covering group SU(2) and thus to the notion of half-integer spin. Given that the universe as it looks is hardly conceivable without half-integer spin particles which all are fermions (according to the spin-statistics theorem of relativistic QFT). Particularly our own existence seems impossible without half-integer fermions as elementary building blocks of matter.