For a fourth provocative slogan about quantum mechanics I’ve chosen:
Use the moment map, not Noether’s Theorem.
Pretty much every physics textbook these days explains the way symmetry principles work as:
- Start with an action functional, invariant under a Lie group G.
- Use Noether’s theorem to get a conserved charge (for each element of the Lie algebra of G).
There’s a short (slightly mystifying) calculation always given to derive this. I’d like to argue that this is really not the best way to think about the implications of having a Lie group act on a physical system, that for this it’s better to take the Hamiltonian point of view. There the way symmetry principles work is:
- For a function on phase space (or on a general symplectic manifold) you get a vector field. This is just Hamilton’s equations, giving the vector field for time evolution corresponding to any Hamiltonian function.
- The infinitesimal action of G on phase space gives a vector field for each element of the Lie algebra of G. The moment map takes an element of the Lie algebra to a function on phase space (the one corresponding to the vector field).
I’m ignoring some subtleties here having to do with the relation between vector fields and functions not being quite one-to-one.
All of the basic examples of conservation laws in physics come about this way. The action of time translation gives the Hamiltonian function, space translation the momentum, rotations give the angular momentum, and phase transformations give charge. You can get these either as moment maps, or using Noether’s theorem.
The moment map however gives you much more, with phase space providing structure that is not visible just from the action. A simple example is the harmonic oscillator in 3 variables. SO(3) rotations act on the configuration variables, preserving the action, so Noether’s theorem gives you 3 conserved quantities, the angular momentum variables. The moment map point of view however gives you much more. The phase space is 6 dimensional (3 positions + 3 momenta) and the Lie group Sp(6,R) of linear symplectic transformations acts on it, with a subgroup U(3) preserving the Hamiltonian. The U(3) includes the SO(3) rotations as a subgroup, but it is much larger (9 dimensions vs. 3), so the moment map gives you many more conserved quantities. After quantization, you learn that energy eigenstates are U(3) representations, telling you much more about them than what angular momentum tells you.
The moment map point of view also gives you quantities corresponding to the directions in Sp(6,R) that are not in U(3). In the quantum theory these act on the full state space (not preserving energy eigenstates) and your state space is a representation of (a double cover of) this group.
For the simplest possible harmonic oscillator, in one-dimension, Noether’s theorem doesn’t really tell you anything. The moment map point of view says that there is an Sp(2,R) acting on phase space, with a U(1) subgroup preserving the Hamiltonian. The moment map is just the Hamiltonian itself. In the quantum theory you find that the harmonic oscillator state space is a representation of (a double cover of) Sp(2,R), with the U(1) action on states characterized by integers, which correspond to the energy. This integrality is the essence of the “quantum” in “quantum mechanics”, and it’s quite invisible to Noether’s theorem, but a basic fact of the moment map point of view.
In some sense this is an argument for the Hamiltonian vs. Lagrangian point of view in general. The relation between the two is that, given a Lagrangian, one constructs a symplectic structure on the space of solutions of the variational problem, and thus a Hamiltonian formalism. Noether’s conserved quantities are then examples of moment maps. The problem is that typically this requires the use of constraints and the quite tricky constrained Hamiltonian formalism.
The positive argument for the Lagrangian point of view is that it comes into its own in the relativistic setting, making Lorentz invariance easy to handle by the Noether’s theorem method. This is quite true, with the standard version of the Hamiltonian formalism distinguishing the time direction and breaking Lorentz invariance. There is however a less well-known “covariant phase space” point of view, where one tries to work with the space of solutions of the equations of motion as one’s phase space. Only if one identifies a solution with its initial data at a fixed time does one distinguish the time direction. I’ve recently enjoyed reading Igor Khavkine’s review article, which in particular does a great job of explaining the history of this line of thinking.
The Lagrangian also comes with the extremely seductive point of view on quantization of the path integral. This point of view works very well for dealing with Yang-Mills theory, and I spent much of my early career convinced that all there was to quantization was figuring out how to make sense of integrating over the exponential of the action. I’m now much more aware of the advantages of the Hamiltonian point of view, especially in terms of understanding quantum theory as representation theory. In some sense what one really wants is to understand quantization in a way that takes advantage of both points of view, but the relationship between them is quite non-trivial.
The discussion here has been far too wordy for most people to make sense of. If you want to understand any of this, you need equations. Luckily, I’ve provided lots of them and many details here, see chapters 12 and 13 [now 13 and 14] for the moment map, chapter 19-22 [now 22 and 24-26] for the harmonic oscillator.
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Hi Peter, I have just discovered your blog.
Question: assuming that quantum information is conserved, what (Noether’s) symmetry would be related to that conservation?
Can the moment map point of view help in answering this?
It is indeed! —at least to this Feynman-naïf, lowly mathematics student. Yours plus the quantum information approach are the only ones so far that have made any remotely intuitive sense to me, without leaving nearly as much of the usual (and inevitable) “shut up and calculate” aftertaste.
Antonio,
I don’t know exactly what “quantum information” is, or the conditions under which it is conserved. From the little I know, I don’t see how either the moment map or Noether’s theorem have anything to do with it since these come into play when you are “quantizing” a classical system, and “quantum information” doesn’t seem to involve that.
To the extent “quantum information” is counting something using integers, typically there’s a U(1) group action around somewhere, explaining why you are seeing integers.
Hi,
Quite interesting your post. One may speculate and say that also Quantum Field Theory should be related to the representation of some group, in the same way that Quantum Mechanics is related with the representation of the Heisenberg group and that also some moment maps should give some conserved quantities. But now this representation space would most probably be the space of distributions or generalized functions and not the nice Hilbert space of $L^2$ functions (in fact it is its closure). And what kind of group would act on such a space? And would it be easer to define a measure on it that it is on a separable Hilbert space?
…just to say that I just got into this:
http://arxiv.org/pdf/hep-th/0206135v1.pdf
Very nice. It will be very helpful, as it has many intersections to what I’m interested in and working on at the moment.
At least for free fields (and, more generally, linear fields in an unquantized background background) qft is based on the Heisenberg group. The difference in QFT is that the dimension of the group is infinite.
The issue of distributions vs. L^2 or some other function space arises long before you get to QFT. Even for a free particle in 1d in QM, momentum eigenstates are not in L^2, and position eigenstates are not functions at all, but distributions. One way I’d like to improve the current version of the manuscript is by doing a better job of dealing with this issue. The problem is that standard ways of doing this (e.g. rigged Hilbert spaces) introduce a lot of complexity even in the very simple basic examples I’m trying to explain.
The new problem in QFT is that the field operators depend on functions (whereas in the QM analog, the position and momentum operators only depend on a finite set of indices).
Yes, but in QM I suppose the use of distributions can be avoided. I even think this was the main motivation of von Neumann in writing his treatise “Mathematical Foundations of Quantum Mechanics” as an alternative, more to the ground of what was known at the time, to Dirac’s description using delta functions. But in QFT as it is a theory of scattering observables have a continuous spectrum, in contrast to QM which is mostly a theory of bounded states, and in that case the commutation relations between conjugate fields are distributional. My intuition is that this must taken into account if one hopes to have a rigorous treatment of QFT.
Joao,
In QM you can also just look at scattering, that’s not really where QFT is different.
The problem in QFT is that your analog of an operator Q_j in QM is a field operator \phi(f), where instead of an index j, the operator depends on a function f. In QM you can multiply and manipulate operators without a huge amount of trouble (although there are problems due to operators being unbounded). In QFT you can do the same, but the problem arises because you want local interactions, so need for example to define not just an operator \phi(x) (the field operator for f a delta-function), but higher-order products of this. That’s where the serious problems come from.
To bring this back to the topic of the posting (or series of postings), my point of view on this is that defining arbitrary local products of field operators is inherently difficult and ridden with ambiguities, with renormalization theory our best wisdom about how to deal with this. If one looks not at arbitrary products, but at ones that correspond to a representation of a Lie algebra, then one gets a much more rigid setup and can perhaps hope to uniquely define such products, or at least better understand exactly what the inescapable ambiguities are.
Unfortunately, this philosophy only now really works for quadratic products, which I try and work out in detail in the notes (the finite-dim versions are complete, still working on the qft part of the notes). Examples like the \phi^4 interacting QFT have no known interpretation of the interaction terms in terms of a representation theory problem. On a more positive note, the SM Hamiltonian (or Lagrangian) only involves quadratic powers of the fermion fields, so, before quantization of gauge fields and the Higgs, the problem doesn’t come up. For pure gauge theory, you do get cubic and quartic terms in the non-abelian case, but there one can perhaps exploit gauge symmetry in some way. For the fully quantized theory of fermi fields coupled to the Higgs and gauge fields, you need some new idea…
Thanks Peter, I have to think about the question I asked you.
I keep your email and in case those moment maps show something interesting, I might contact you again.
What you said reminded me of the issue of defining a product of distributions that Martin Hairer had to deal with in his theory which he applied to Φ^4_3 Euclidean quantum field theory. It also involves some form of renormalization.
Bill,
Yes, I think this is essentially the same problem Hairer is dealing with. I see that in his paper he writes
“the mathematical analysis of QFT was one of the main inspirations in the development of the techniques and notations presented in Sections 8 and 10.”
As far as I know the major successes of Quantum Mechanics in describing the real world are the exact solutions of the Hydrogen atom and the Harmonic Oscillator, which are bound states and as so have a phase space that is compact. And when you consider differential operators on a compact manifold you get a discrete spectrum and a countable base of eigenfunctions and so a separable Hilbert space is natural. Of course, formally you can consider the eigenvalue equation for the momentum operator on R and this gives plane waves for eigenfuntions and continuous eigenvalues, but does this describe anything in Quantum Mechanics beside a free particle?
Maybe you have a different perspective, but in my view this is one of the major differences between QFT and QM, the fact that in QFT generalized functions are unavoidable.
Joao,
A large part of QM is scattering theory, and to do this you have to handle the continuous spectrum (and distributions are a good way to do it).
Again, if you look at the hard problems of QFT, they’re due to the nature of interactions as local products of quantum fields. Yes, part of this story is that of problems with defining products of distributions, but it’s best to understand what the fundamental nature of the problem is.
Hi, Peter. I’m glad you enjoyed the little historical overview in my article of the ideas leading to a covariant view of phase space. As with many deep ideas, its history is a much tangled web, which I find fascinating. Thanks for the mention, btw!
OK, Peter, it has been an interesting discussion. Thank you for replying.
Regards.
Dear Peter, regarding the best mathematical framework for QFT, I favour the C*-algebra point of view, which includes the group perspective you take,
i.e. is more general, and has a well controlled representation theory.
(1) Regarding free bosons;- here one takes the Weyl C*-algebra which is a twisted discrete group algebra of the underlying symplectic space. It gives you those representations of the associated Heisenberg group where the central element maps to the identity, which is what physics wants. The “momentum eigenstates” you mention, correspond to certain states which are nonregular (i.e. they are discontinuous on the underlying Heisenberg group), but which are quite well-defined (see Verbeure on plane waves). So rigged Hilbert spaces are unnecessary. Other C*-algebras are also possible, e.g. the C*-algebra generated by the resolvents of the smeared fields.
(2) For free fermions, you need to take the CAR-algebra, which does not fit well into your group perspective, but it is a very well-behaved C*-algebra. If you have physical symmetry groups acting on the fermions, you will take appropriate crossed products, to get covariant representations.
(3) Renormalization (at least in lattice C*-models) can be understood as a procedure which moves you out of one representation into another, which is why the perturbation series cannot converge. But at the C*-level it makes sense.
(4) I agree that pointwise products of the fields (Wick products) are hard to understand – though have been rigorously constructed in the free field case. These currently do not fit well into the C*-algebra picture.
Regards
Hendrik,
Thanks. The C* algebra point of view is deeply related to representation theory. For fermions I try to make clear in the notes that there is a perfect parallelism between the symplectic/Heisenberg story for bosons and the orthogonal/Clifford story for fermions. To the extent C* algebras don’t equally well handle either case, that’s a problem
Thanks Igor, your paper was quite enlightening. The question of the relation of the Hamiltonian and Lagrangian viewpoints seems to me surprisingly still not completely satisfactorily understood.
Hi Peter,
In symplectic geometry, people often require moment maps to be equivariant (wrt the co-adjoint action), which is needed for things like Kirwan convexity to hold. Non-equivariant moment maps are nevertheless also useful in some contexts. Could you comment on the role equivariance plays for your purposes?
Hi Johan,
I’m generally assuming that if possible the moment map (which is only defined up to a constant) is chosen to be equivariant, so you have a Lie algebra homomorphism from the Lie algebra to functions on phase space, which becomes a Lie algebra representation when you quantize.
When the Lie algebra has non-trivial central extensions (H^2 non-zero), then you will have non-equivariant moment maps that can’t be made equivariant. This is what physicists call the “anomaly”. It’s usually thought of as a purely quantum effect, but you do actually see it this way even at the classical level.
The finite-dimensional groups I’m writing about in the notes all have vanishing H^2, so one can take the moment map to be equivariant. There are some comments there about the situation in infinite-dimensions, but that’s mostly beyond the scope of what I’m trying to write about there.
Not sure if this addresses what you’re thinking about. Quite likely you know about some interesting examples of use of non-equivariant moment maps that I’m just unaware of.
The kind of anomaly given by obstructions against lifts from actions by Hamiltonian vector field to the Poisson bracket Lie algebra is typically called a classical anomaly (e.g. Arnold’s book, appendix 5.A).
Every finite-dimensional Lie algebra that can be embedded into gl(N) for some N can be realized as a function on phase space. Obviously, since E^i_j = q^i p_j generate gl(N) under the Poisson bracket. Classically the same is true in infinite dimensions, but quantization leads to infinitites and is problematic.
What is perhaps not so well-known, but obvious once you think about it, is that most infinite-dimensional Lie algebras of interest in physics can also be realized as functions on a *finite*-dimensional phase space. Namely, the functions f^i(q) p_i generate the algebra vect(N) of vector fields, so every algebra that can be embedded into that can be realized as functions over phase space. Since the phase space is finite-dimensional, quantization is not a problem.
This is a kind of first-quantized approach and not directly relevant to physics, but a slight variation of this theme actually yields interesting representations of infinite-dimensional Lie algebras.
Famously, the momentum map for the action of the Galilean group G on the phase space of a non-relativistic particle (or N such particles) cannot be chosen to be equivariant, as long as the action of G on its dual Lie algebra g* is the usual coadjoint action. On the other hand, a group cocycle of G with coefficients in the coadjoint representation can be used to modify the action of G on g* and make it equivariant (as discussed, for instance in these slides by Charles-Michel Marle). The same cocycle (or rather its infinitesimal version) shows up in the fact that the the boost and translation generators cannot be chosen such that their Poisson bracket is zero (even though their Hamiltonian vector fields commute). Instead, their Poisson bracket is a constant proportional to the total mass of the N-particle system. This is indeed a famous example of a classical anomaly.
Peter,
This is a quick not very well thought out reply.
The symmetries of that you are talking about are manifest if one uses the
Schwinger-Keldysh formulation where the fields are doubled. This is
useful for discussing real time physics at finite temperature.
For instance the action of the harmonic oscillator is
S \sim \int dt (\dot x_1)^2 – x_1^2 – ((dot x_2)^2 – x_2^2)
You might want want to think about it in these terms. The Keldysh
setup is most useful close to the classical limit. See for example,
hep-ph/0212198