Lie Groups, Lie Algebras, Exp Maps & Unitary Ops in QM

[quasar_4]

Can anyone expand on the relationship between Lie groups, Lie algebras, exponential maps and unitary operators in QM? I've been reading lately about Lie groups and exponential maps, and now I'm trying to tie it all together relating it back to QM. I guess I'm trying to make sense of how Lie groups emerge in QM, and also how the physics approach to Lie groups (that I've seen so far, at least) tends to emphasize the fact that a subset of the Lie group in the neighborhood of the identity can generate the entire group (rather than thinking about Lie groups strictly as groups endowed with a C^infinity manifold). I'm not totally sure why all this becomes so relevant in QM - anyone care to comment? :)

 

[RedX]

Every representation of a group is equivalent to a unitary representation, so you can forget about non-unitary ones and just concentrate on unitary ones which are nicer. I vaguely recall the proof of this, and if I recall correctly, this holds for all groups, and not just Lie groups. So every representation of a group is equivalent to a unitary representation.

The Lie algebra is basically the multiplication table for the Lie group. Once you know the Lie Algebra, you know how to multiply group elements. The formula you would use is called the Baker-Campbell-Hausdorff formula, which expresses the multiplication of two group elements in terms of the commutator of the generators.

Transformations that can be generated from many infinitismal transformations are common in physics. Putting all the infinitismal transformations together you get the exponential. This is how Lie groups come in, from a bunch of infinitismal transformations combining to a finite transformation.

 

[schieghoven]

However, different groups can have the same algebra; SO(3) and SU(2) are the standard example.

 

[Fredrik]

The Dirac-von Neumann axioms define the general framework of QM, but to define a specific theory of matter in that framework, we need something more. Theories of non-interacting matter can (almost) be specified by choosing a symmetry group that includes translations in time. Choose the Galilei group and you end up with wave mechanics (i.e. wavefunctions, the Schrödinger equation and all that stuff). Choose the Poincaré group and you end up with relativistic quantum mechanics.

Two physical observers related by a Lorentz transformation g use different mathematical representations of the same state. (If you say "it's on my right", a rotated observer might say "it's in front of me"). If we want both of these guys to be able to predict probabilities of possible results of experiments, regardless of what g is, then we must assume that for each g in the group, there exists a symmetry T(g) on the set of unit rays. A symmetry is a bijection that preserves probabilities in the sense that if $|\psi_1\rangle$ and $|\psi_2\rangle$ belong to the rays R₁ and R₂ respectively, we have $|\langle \psi_1|\psi_2\rangle|^2=|\langle \psi_1'|\psi_2'\rangle|^2$ for all $|\psi_1'\rangle$ and $|\psi_2'\rangle$ belonging to the rays R₂'=T(g)R₁ and R₂'=T(g)R₂ respectively. A theorem proved by Wigner guarantees that for each g, there also exists an operator U(g) on the set of state vectors, that's either linear and unitary or antilinear and antiunitary. This is where it's significant that the group is a Lie group. I think the theorem assumes that we're dealing with a simply connected Lie group. That's why the proper orthochronous subgroup of the Lorentz group is so important.

The map U isn't quite a group homomorphism. Instead of U(gg')=U(g)U(g'), we have U(gg')=C(g,g')U(g)U(g'), where C(g,g') is a phase factor. There are some tricks we can use to get rid of the phase factor. The crucial step is to replace the rotation group SO(3) with its covering group SU(2). This doesn't change the physics significantly. So now we can deal with a unitary representation of a simply connected Lie group. Remarkably, these ideas lead directly to the concept of elementary particles. Each particle species corresponds to an irreducible representation. Check out chapter 2 in Weinberg for more details on this.

What remains to be specified in order to complete the theory is an identification of operators with observables. For example, when our g is a translation in time, the corresponding unitary operator can be expressed as exp(-iHt), and we identify this H with the observable "energy". These identifications can be partially justified, but I think we still have to consider them axioms.

What about quantum fields? We clearly don't need quantum fields to define particles, so what are they good for? They are used to specify interactions between particles. The non-interacting quantum field theories (theories derived from Lagrangians with no higher powers of the field components than the quadradic term) can also be used to explicitly construct irreducible representations and Hilbert spaces of one-particle states, which can then be used to construct the Fock space of state vectors of arbitrary numbers of non-interacting particles.

The above is just one of the two most important applications of Lie groups in particle physics. The other one is more difficult, I think. It involves considering certain fiber bundles over spacetime to get a nice mathematical framwork for gauge theories. I'm afraid I can't describe the details very well since I'm still trying to learn that stuff.

 

[meopemuk]

The Dirac-von Neumann axioms define the general framework of QM, but to define a specific theory of matter in that framework, we need something more. Theories of non-interacting matter can (almost) be specified by choosing a symmetry group that includes translations in time. Choose the Galilei group and you end up with wave mechanics (i.e. wavefunctions, the Schrödinger equation and all that stuff). Choose the Poincaré group and you end up with relativistic quantum mechanics.

Two physical observers related by a Lorentz transformation g use different mathematical representations of the same state. (If you say "it's on my right", a rotated observer might say "it's in front of me"). If we want both of these guys to be able to predict probabilities of possible results of experiments, regardless of what g is, then we must assume that for each g in the group, there exists a symmetry T(g) on the set of unit rays. A symmetry is a bijection that preserves probabilities in the sense that if $|\psi_1\rangle$ and $|\psi_2\rangle$ belong to the rays R₁ and R₂ respectively, we have $|\langle \psi_1|\psi_2\rangle|^2=|\langle \psi_1'|\psi_2'\rangle|^2$ for all $|\psi_1'\rangle$ and $|\psi_2'\rangle$ belonging to the rays R₂'=T(g)R₁ and R₂'=T(g)R₂ respectively. A theorem proved by Wigner guarantees that for each g, there also exists an operator U(g) on the set of state vectors, that's either linear and unitary or antilinear and antiunitary. This is where it's significant that the group is a Lie group. I think the theorem assumes that we're dealing with a simply connected Lie group. That's why the proper orthochronous subgroup of the Lorentz group is so important.

The map U isn't quite a group homomorphism. Instead of U(gg')=U(g)U(g'), we have U(gg')=C(g,g')U(g)U(g'), where C(g,g') is a phase factor. There are some tricks we can use to get rid of the phase factor. The crucial step is to replace the rotation group SO(3) with its covering group SU(2). This doesn't change the physics significantly. So now we can deal with a unitary representation of a simply connected Lie group. Remarkably, these ideas lead directly to the concept of elementary particles. Each particle species corresponds to an irreducible representation. Check out chapter 2 in Weinberg for more details on this.

What remains to be specified in order to complete the theory is an identification of operators with observables. For example, when our g is a translation in time, the corresponding unitary operator can be expressed as exp(-iHt), and we identify this H with the observable "energy". These identifications can be partially justified, but I think we still have to consider them axioms.

What about quantum fields? We clearly don't need quantum fields to define particles, so what are they good for? They are used to specify interactions between particles. The non-interacting quantum field theories (theories derived from Lagrangians with no higher powers of the field components than the quadradic term) can also be used to explicitly construct irreducible representations and Hilbert spaces of one-particle states, which can then be used to construct the Fock space of state vectors of arbitrary numbers of non-interacting particles.

Hi Fredrik,

this is a very nice summary of the Poincare Lie group applications in quantum theory. I just wanted to point out one more non-trivial (and under-appreciated) consequence of the Poincare symmetry (first noticed by Dirac and discussed in Weinberg's book): This symmetry requires that relativistic interaction terms cannot be added only to the Hamiltonian (the generator of time translations). There should be also interaction terms in the generators of boosts. This may indicate that "Lorentz transformations" in interacting systems do not have the simple linear form known from special relativity.