It’s getting late, but I can’t help myself. Reading too many wrong things about symmetry and physics on Twitter has forced me to do this. And, John Baez says I don’t explain things. So, here’s what the relationship between symmetry and physics really is.
In the language of mathematicians, talking about “symmetries” means you are talking about groups (often Lie groups, or their infinitesimal versions, Lie algebras) and representations. The relation to physics is:
Classical mechanics (Hamiltonian form)
In classical mechanics the state of a system with $n$ degrees of freedom is given by a point in phase space $P=\mathbf R^{2n}$ with $n$ position coordinates $q_j$ and $n$ momentum coordinates $p_j$. Functions on this space are a Lie algebra, with Lie bracket the Poisson bracket
$\{f,g\}$. Dynamics is given by choosing a distinguished function, the Hamiltonian $h$. Then the value of any function on $P$ evolves in time according to
$$\frac {df}{dt}=\{f,h\}$$
The Hamiltonian $h$ generates the action of time translations. Applying the same formula, other functions generate the action of other groups (spatial translations, rotations, etc.). If your function satisfies $\{f,h\}=0$, it generates a “symmetry”, and doesn’t change with time (is a conserved quantity).
Quantum mechanics
Quantization of a classical system is something mathematically obvious: go from the above Lie algebra to a unitary representation of the Lie algebra. This takes elements of the Lie algebra (functions on $P$) to skew-adjoint operators on a Hilbert space, the space of quantum states. There’s a theorem (Stone-von Neumann) that says that (modulo technicalities) there’s only one way to do this, and it gives an irreducible unitary representation that works for polynomials up to degree two. For higher degree polynomials there will always be “operator ordering ambiguities”. The representation is given by
$$1\rightarrow -i\mathbf 1,\ \ q_j\rightarrow -iQ_j,\ \ p_j\rightarrow -iP_j$$
This is a representation because
$$\{q_j,p_k\}=\delta_{jk}\rightarrow [-iQ_j,-iP_k]=-i\delta_{jk}\mathbf 1$$
The right-hand side is the Heisenberg commutation relations for $\hbar=1$.
For more details, I wrote a whole book about this.
Hi Peter, old student from your Lie groups class in the early 2010’s here. (Though in hindsight, I wish I had taken your math physics class instead). Here’s an easy way to complete your post with the last piece of the “holy trinity” of math physics: relativistic mechanics.
Given the $(n+1)$-dimensional Minkowski spacetime with the standard Minkowski metric $(\mathbb{R}^{n+1},m)$, you can consider the Poincaré group; the group of symmetries of m. These symmetries include spatial and temporal translations, rotations, and boosts. By Noether’s theorem, these correspond to several conserved quantities to solutions of the linear wave equation $\Box_m \phi = -\partial_t^2 \phi + \Delta_x \phi = 0$. These symmetries have paved the way for the dramatic progress in mathematical relativity seen in the past 3 decades.
Your blog is great as it is. Never mind twitter. There are actually only very few frequently updated math and physics blogs which succeed explaining non-obvious things for a larger audience (some linked here on your site).
Leonardo Abbrescia,
The point of view on symmetry I describe can handle the symmetries of relativistic mechanics, you just need to think of the phase space as the space of solutions of your relativistic wave equation. It doesn’t tell you why certain groups and representations occur in our best fundamental theories (or why a certain wave equation). It does tell you rather that classical and quantum mechanics are formalisms whose structure is based on the use of a specific infinite dimensional Lie algebra.
This point of view is very different than the one normally taught to physicists, that it’s all about Noether’s theorem (which, when you have a Lagrangian with an invariance group, tells you the functions that generate the symmetry) and just a calculational method useful in certain approximations.
oliver knill,
Thanks! John Baez is however right that normally writing expository material is not what I do on the blog (and he’s one of very few who does a wonderful job of this on the internet). There are several reasons I don’t do this, the biggest one that it’s really hard.
Great summary. Who needs a book! 🙂
Indeed it was intended as a description, not a criticism. Yours is one of the few blogs I follow.
You may be interested in recent work on categorical and non-invertible symmetries, those don’t have a group structure.
Moshe,
Thanks. I want to emphasize though that this posting is not about a modern, sophisticated new development in ideas about symmetry. It’s about the simplest topics at the very beginning of first courses in Hamiltonian mechanics and QM (the Poisson bracket and the Heisenberg commutation relations).
That the Poisson bracket is a Lie bracket goes back to the very definition of and motivation for the Lie bracket 150 years ago, and the idea that QM gives a unitary representation of the classical mechanical Lie algebra goes back 100 years to Dirac’s major insight at the very beginning of QM. Dirac would not have used this language, but Weyl immediately was undoubtedly aware of it.
This line of thought is developed until its final logical consequences in George Mackey’s work on Imprimitivity Systems, which unifies the Stone-von Neumann Theorem (seen as the uniqueness of the irepresentation of the Heisenberg group/Weyl relations up to unitary eq.) and Wigner’s work (on the classification of elementary particles via irepresentations of the Poincaré group) into a single mathematical framework. Furthermore, it also clarifies (at least for the finite dimensional phase space case) the origin of Dirac’s quantization rule (Poisson goes to i×operator brackets.)
Really beautiful mathematics, I wish it was more well known.
Standard references can be:
-Folland: Introduction to Harmonic Analysis;
-Varadarajan: Geometry of Quantum Theory*.
*which also contains a proof of Gleason’s Theorem and the whole lattice theoretical approach to QM.
Modern developments include Landsman’s work on translating and expanding Mackey’s ideas into the C*-algebra approach, aided by the use of tools from noncommutative geometry.
Alex,
The “line of thought” I’m explaining here is just very basic facts about a specific Lie algebra and a specific representation, with the significance that these are basic postulates of classical and quantum mechanics respectively.
You can develop this “line of thought” in many different directions, bringing in more groups and representations. Mackey’s theory is only one specific direction, aimed at a general theory of induced representations.
Personally I’d argue that a generalization more relevant to physics is in the direction of geometric representation theory, the orbit method and geometric quantization. These are topics which link symplectic geometry (more general notions of classical phase space) and representation theory (more general notions of quantum system). This is a genuinely mathematically sophisticated and active area of current research, tough going for most people.
I think it’s a mistake to throw a lot of sophisticated abstract formalism designed to handle very general questions at people as an “explanation” of the role of symmetry in physics. Hardly anyone will understand this, and all you’ll accomplish is convincing physicists that this area of mathematics is the “gruppenpest”, a plague of incomprehensible ideas that tell us nothing about physics. Effort would be better spent explaining to physicists that Lie algebras and representations provide real insight into basic formulas they were taught to accept unthinkingly about classical and quantum mechanics
Peter,
yes, Mackey’s direction is one among several, but, historically, is the closest one (early 40s) to the original developments (Weyl, early 30s; Wigner late 30s, etc.), and was devised as a synthesis of all of those developments that were happening in QM at the moment. That’s why I used the term “line of thought”, since it seemed like a natural and organic development to me. In fact, it was Mackey who first used the name “Stone-von Neumann Theorem” for the result about the uniqueness of the CCRs.
Geometric quantization is indeed another approach, and I think they are actually complementary, since they treat different problems and aspects of quantization. In fact, they can even be combined. Unfortunately, both approaches have their well known limitations. Geometric quantization uses the language of connections and bundles; the hard part, I think, lies not on this but on the amount of tricks and subtle twists you must do along the way in order to make it work. Due to this, the process can become quite artisanal and difficult to follow for the unexperienced.
As for the clarification part, I was referring to the simple fact that the imprimitivity condition (which can be seen as a simple covariance condition on the system) gives rise to the CCRs.
Finally, I actually disagree with your last paragraph. In my own experience, I have seen many graduate students very confused by all the seemingly unrelated Lie algebras/groups that were introduced in their courses (Galilei, Heisenberg, Poincaré) and were quite interested and relieved when told that the representation theory of all of these groups could be solved by the application of a single general theorem, and by many other aspects as well (e.g., obtaining the wave equations from the induced representations’ spaces, etc.) I do agree with the part about explaining the role of Lie algebras in the formulas that are thrown at them in the basic courses, and it was never my intention to suggest replacing that by some full blown introduction to induced representations, geometric quantization or whatever. Just an after that elaboration.
Alex,
For me the imprimitivity condition doesn’t clarify the CCRs, they (the CCRs) are just the Lie algebra relations for linear functions on phase space.
A lot of the appeal of representation theory to me is that it’s a vast subject deeply connected to physics that one can approach from many different points of view. There may very well be an overarching single point of view that unifies the subject, but that remains elusive (at least to me…)
Peter,
well, the idea is that the imprimitivity condition is a covariance-type condition, between the position observables on some configuration space and the relevant group action on it, which in simple terms can be viewed as expressing the homogeneity of said observables with respect to the relevant symmetries of the configuration space. Furthermore, in the standard examples where the configuration space is R^n and the group are the additive translations R^n, the imprimitivity condition is equivalent to the Weyl relations(/Heisenberg group rep.), which are just the exponentiated form of the CCRs: thus, in this view, the CCRs are equivalent to imposing the homogeneity of the position observables with respect to the relevant symmetry of the configuration space. Thus, it’s a very symmetry-friendly explanation, that’s why I considered it relevant here.
I don’t claim Mackey’s theory to be such an overarching single point of view on representation theory, but it does unifies, as I mentioned, the Wigner classification theory (giving also the representation spaces, made of -distributional- solutions to the relevant wave equations for all spins in full mathematical rigor*) with the CCRs and their uniqueness (it can also be used to obtain the relativistic position operators and show that they indeed don’t exist in the massless case for spin >=1, like the photon and graviton).
*most students of QFT think that single photons don’t have wave functions and equations because that’s what they are told in informal presentations of the subject. But they do have! And are given by the elements of the corresponding representation spaces, of which, as far as I know, the Mackey machine method via induced representations is one of the few approaches if not the only one to give you those Hilbert spaces rigorously and explicitly only from Poincaré representation theory. To construct the Fock space for photons in standard QFT, you need first such single photon spaces. In mathematical physics, these details are indeed relevant. If physicists don’t care, that’s their loss, not mine.
One can define both in terms of analysis instead of Algebra, with symmetries playing an important but auxiliary rather than central/conceptual role:
Classical mechanics: There is an initial condition and an action. Dynamics is defined by the trajectory that minimizes the action, which gives evolution at any point after the initial condition.
Quantum mechanics: Same as classical mechanics, but dynamics is defined by correlators that are computed in terms of functional integrals of trajectories.
It is not clear, at least to me, why the symmetry/algebra definition is “better” or “more fundamental” than the analysis based one.
“*most students of QFT think that single photons don’t have wave functions and equations because that’s what they are told in informal presentations of the subject. But they do have!”
Of course they do. But you just need to write the wave equation (for any relativistic particles, not only photons) in terms of oscillator variables (which become creation and annihilation operators in second quantization). These variables are the Fourier transform of the true wave function. Amplitudes and probabilities become as simple as they are for the Schroedinger equation, No deep mathematics is needed to understand this.
Since the post advertises that it will explain “what the relationship between symmetry and physics really is,” I think it’s important to add at least a caveat that discrete symmetries are also important in physics (especially condensed matter physics) and are not described by a Lie group.
Gavin,
Yes, and I did start off the post referring to all groups, not just Lie groups. In classical mechanics discrete groups are of limited use, no Noether’s theorem. In quantum mechanics however, the abstract situation is the same as for Lie groups: the state space is a unitary representation of the group, and the representation theory of the discrete group then provides a lot of information about the behavior of the quantum system.
lun,
Yes, if you want you can do physics purely within the Lagrangian formalism (although if you do this, you quickly find that “just integrate $e^{iS}$ over the infinite dimensional space of trajectories” often is a proposal that one can’t make well-defined).
The question was about the relation of symmetry and physics, and I’m arguing that is best understood in the Hamiltonian formalism (compare solving the N-dim harmonic oscillator problem and studying its symmetries in the canonical operator formalism vs. by calculating path integrals).
Alex/Peter Orland,
This is getting off topic. I’ll just again refer to my book, where the single-particle state space for relativistic particles is worked out (even including distributional solutions, although that is not handled well and I hope to rewrite some sections for that reason).
I’m missing distinction between symmetries and invariances. They are not the same concepts, although both correspond to some transformations that leave the theory intact. Symmetries (e.g. rotational symmetry) map some physical states to different physical states. Invariances (e.g. gauge, diffeomorphism) do not change the physical state. As Coleman used to say: You don’t ask the experimentalist “In what gauge you did your experiment?”.
maciej,
This is a good point. When I was writing the book about QM, I struggled with the fact that it’s not clear what one means when one says “symmetry”. There’s always a group, and it is acting on something, but what structure is being preserved has to be specified and so different sorts of things can be going on.
In the Hamiltonian formalism, functions on phase space with Poisson bracket are the Lie algebra for the infinite dimensional group of transformations of phase space preserving the symplectic structure. This is a group of “symmetries”, but sometimes by “symmetry” one means something more special: a Lie algebra of functions that have zero Poisson bracket with the Hamiltonian. These also preserve time evolution, give conserved quantities. As an example, consider spatial rotations: components of angular momentum give a three-dimensional space of functions on phase space with Poisson brackets satisfying the relations of the Lie algebra of the rotation group. These functions may or may not Poisson commute with the Hamiltonian and be conserved quantities (depending on whether the Hamiltonian is invariant under all rotations, or just a subgroup).
In QM, you have the same issue, you get operators with commutation relations given by the Lie algebra, which may or may not commute with the Hamiltonian operator. But in QM you in addition have a representation of these operators on states. Your states may be invariant (trivial representation), or in some specified non-trivial representation. In the rotation group example, your states may be invariant under rotations (s-wave), or transform non-trivially under rotations in a manner given by the representation theory of SO(3) and SU(2).
In the case of the gauge group, conventionally you assume states are in the trivial representation for local transformations (not necessarily so for global transformations).
I am not sure I agree with what you say about the transition from classical to quantum. Real classical variables become hermitian operators (without any factors of i)
And the classical Poisson bracket is then replaced -i times the operator commutator,
with 1/hbar if this is not set equal to one. This is what Dirac did in 1926.
The centenary of real QM is due in just three years.
Hugh Osborn
Hugh Osborn,
In a unitary Lie algebra representation the operators are skew-adjoint, not self-adjoint, so I put in factors of i to change the usual self-adjoint operators to skew-adjoint. Using these skew adjoint operators, the Lie bracket of classical mechanics (the Poisson bracket) goes precisely to the commutator of operators (this is the definition of a Lie algebra representation).
If you write things just in terms of self-adjoint operators, you have the usual extra factor of i in the relation between the Poisson bracket and the commutator of operators.
While group theory (more exactly representation theory of Lie Groups) is certainly important in Quantum Field and Condensed Matter theories, it is hard to find any discussion of it in the literature of the rest of physics. What books are there on Classical Mechanics and Classical Electrodynamics from the group theory point of view? Thanks in advance.
Ricardo Jimenez,
I just noticed this book
https://www.amazon.com/dp/1848167741
but don’t know much about it. I don’t know of any book that develops classical mechanics from the point of view of taking as basic the Lie algebra of functions on phase space with Lie bracket the Poisson bracket.
For classical electrodynamics, the relevant group theory is the Lorentz group and the gauge group, I don’t know of a book that puts those front and center but maybe there is one.
In classical physics you don’t really encounter representation theory much, although that’s fundamental to quantum physics. So it is in quantum physics where the Lie group point of view really comes into its own.
Wouldn’t classical mechanics texts more in the direction of symplectic geometry cover that sort of material?
Jonathan,
I took a look at one such text I could think of (Arnold). His and other books do emphasize that phase space is a symplectic manifold and use the symplectic structure, but don’t emphasize that the Poisson bracket is Lie bracket for the Lie algebra of the infinite dimensional group of transformations of phase space preserving the symplectic structure (maybe this fact is somewhere, but not emphasized).
The book “Introduction to Mechanics and Symmetry” by J.E. Marsden and T.S. Ratiu contains many interesting applications of Lie groups and Lie algebras in Classical Mechanics.
Alex,
”Geometric quantization is indeed another approach, and I think they are actually complementary, since they treat different problems and aspects of quantization. In fact, they can even be combined. Unfortunately, both approaches have their well known limitations.”
In my recent paper
A. Neumaier and A. Ghaani Farashahi, Introduction to coherent quantization, Analysis and Mathematical Physics 12 (2022), 1-47.
I discuss coherent quantization, a generalization of geometric quantization including the non-unitary case. Unlike in geometric quantization, the groups are not assumed to be compact, locally compact, or finite-dimensional. This implies that the setting can be successfully applied to quantum field theory, where the groups involved satisfy none of these properties.
Ricardo Jimenez,
”What books are there on Classical Mechanics and Classical Electrodynamics from the group theory point of view?”
Two nice such books include:
Marsden and Ratiu, Introduction to mechanics and symmetry.
Arnold and Khesin, Topological methods in hydrodynamics.
I’ve always felt that Ballentine’s Chapter 3 in his Quantum Mechanics books is a tour de force. In one chapter he introduces the Gallilean group as the group that describes the non-relativistic symmetries of space time and proceeds to derive the fundamental Quantum Mechanic operators from these symmetries.
He doesn’t get into the formal mathematics of Lie Groups and Algebras or representations but it was a powerful illustration of that POV.
I’d contrast that with your book Peter, which I have tried to work through on and off, which goes much deeper into the subject, but, if I may offer a slight critique, the motivation for the approach is less clear.
Student,
My book does have a somewhat similar emphasis on non-relativistic space-time symmetries, but sticks to the Euclidean group of rotations and translations rather than getting into the larger Gallilean group (for which the representation theory is much trickier, and the extra generators you get not that interesting). In both cases, thinking of the non-relativistic case this way extends nicely to the relativistic Poincare case.
One thing to keep in mind is that Ballentine is aimed at physicists, my book is based on a course in the math department. The motivation is much more on the math side than for Ballentine. A book like Ballentine’s is hard for mathematicians to follow, since he’s avoiding standard mathematical concepts (like that of a representation) in favor of language familiar to physicists. For physicists, I suspect my book is similarly harder to follow because of the unfamiliar language and formalism, but I hope that they will ultimately find this point of view worth learning, since it is quite powerful and clarifies issues that the physicist’s approach often leaves unclear.