Some preliminary categorical remarks

Let be a ring. You probably know that a ring is a set with two operations, addition and multiplication, satisfying some compatibility relations. I prefer the following definition: a ring is a monoid object in the monoidal category of abelian groups under tensor product. This definition makes it clear that rings are “linearized” monoids and emphasizes that rings naturally appear as the endomorphism rings of abelian groups, just as monoids naturally appear as the endomorphism monoids of sets.

Another way to state this definition, perhaps even more categorical, is that a ring is a small -enriched category with one object, just as a monoid is a small category with one object. Among other things, this definition encourages us to think of -enriched categories (such as abelian categories) as “rings with many objects” or ringoids, named in analogy to groupoids. It also shows that the construction of the opposite ring is a special case of the construction of the opposite category.

Definitions and examples

A left -module is an abelian group equipped with a ring homomorphism . Equivalently, it is equipped with a map satisfying certain properties, hence the term “left”; right -modules are defined similarly except with a map (or equivalently a homomorphism ). Categorically, thinking of as a category as above, a left -module is an enriched functor (and a right -module is an enriched contravariant functor). Without qualification, “module” and “-module” mean “left -module.”

A morphism of modules is an abelian group homomorphism respecting the action of (equivalently, it is an enriched natural transformation – okay, I’ll stop now). A submodule of is a subgroup preserved by the action of . Modules admit obvious notions of direct sum, kernel of a morphism, and quotient by a submodule. A module is

• simple if it is nonzero and its only submodules are and (equivalently, if its only quotient modules are and ),
• indecomposable if it cannot be written as a direct sum of two nonzero submodules, and
• cyclic if it is generated as a module by a single element (equivalently, if it is a quotient of as a left -module).

Proposition: Any simple module is cyclic.

Proof. Let be simple and a nonzero element of it. Then the map has image a nonzero submodule which must therefore be all of .

Example. Let be a field. A -module is just a -vector space. The simple -modules are precisely the one-dimensional vector spaces.

Example. Let be a polynomial ring. A -module is a -vector space together with a linear operator acting on it. The simple -modules are precisely the one-dimensional vector spaces where acts by a scalar and the finitely-generated indecomposable -modules are precisely the vector spaces where acts by a Jordan block by the theory of Jordan normal form. This example shows that an indecomposable module need not be simple.

Example. Let be the free -algebra on two generators, or the ring of noncommutative polynomials in two variables. An -module is a -vector space together with two linear operators acting on it, and the classification of -modules is such a difficult problem that it gave rise to the notion of a wild classification problem; see also this MO question.

Example. Let be a small category. The category of functors is naturally equivalent to the category of left -modules for the category algebra of . This is the algebra generated by the morphisms of , where the composition of two morphisms is their composition in if defined and zero otherwise.

This construction is, of course, of considerable interest when is a group regarded as a one-object category; it (or for a field) gives the group algebra. It is also of considerable interest when is a free category on a graph , in which case (or for a field) is called the path algebra or quiver algebra of the quiver .

Simple modules

The example of above shows that not every module is a direct sum of simple modules. Nevertheless, it would be nice to understand general modules in terms of simple modules since their behavior is, well, relatively simple. For example, the following is true.

Schur’s lemma: Let be two simple -modules. Then is empty if are non-isomorphic; otherwise, is a division ring .

Proof. Let be a morphism. Then the kernel and image of are submodules of respectively, which must by hypothesis be empty or the entire module. So either or is an isomorphism.

Example. Let be the real group algebra of a finite group . Then an -module is a real representation of . The division rings that can arise as endomorphism rings of simple -modules are necessarily finite-dimensional division rings over , hence by the Frobenius theorem can only be , or the quaternions , and all three examples occur. See Frobenius-Schur indicator.

What can we say about understanding a general module in terms of simple modules? One naive idea is to take an arbitrary module , find a simple submodule of it, consider the quotient by that submodule, find a simple submodule of that, and so forth. Unfortunately, it is false that every module contains a simple submodule! For example, consider acting on itself by left multiplication. The simple -modules are the finite cyclic groups of prime order, which don’t appear as submodules of .

Rather than submodules, we need to think about quotients. By Zorn’s lemma, any module has a maximal proper submodule , and by maximality the quotient is necessarily simple. We can iterate this construction with the submodule , but it is not guaranteed to terminate (indeed in the example of above it cannot terminate). If it does – that is, if we can find a composition series

of submodules such that the composition factors are simple, we say that has finite length. Such a module is, in an appropriate sense, built up from the simple modules , but if we are to take this idea seriously it would be nice if this list of modules did not depend on the choice of composition series. This is easy to see concretely for : here the finite length modules are precisely the finite-dimensional ones, and the list of simple modules can be identified with the list of eigenvalues of , which only depends on the module. So it is not totally unreasonable to hope that this is true generally.

Theorem (Jordan–Hölder): Let be a finite-length module. Any two composition series have the same length , the length of , and the composition factors appearing in them are the same up to permutation.

Proof. Let and be two composition series. We induct on . The statement is obvious if . In general, if then the statement follows by the inductive hypothesis. Otherwise, by maximality , hence

by the isomorphism theorems. WLOG ; then intersecting by and discarding terms, we can find a composition series for of length at most , and attaching such a composition series to the beginnings of the series

we find two series which clearly have the same length and the same composition series. On the other hand, by the inductive hypothesis the first series is equivalent to and the second series is equivalent to . The conclusion follows.

Semisimple modules

A module is semisimple if any submodule of is a direct summand. This turns out to be a particularly nice condition to work with on modules. For example, it is true of all simple modules, but it is also closed under direct sums, quotients, and taking submodules. In particular, every direct sum of simple modules is semisimple. What can we say about other semisimple modules?

Proposition: Any semisimple module contains a simple submodule.

Proof. Let be nonzero. It suffices to reduce to the case that . By Zorn’s lemma, there is a submodule of maximal with respect to the property of not containing . By assumption, there is a direct sum decomposition

.

If contains a nonzero submodule , then by maximality contains , hence , so . Hence is a simple submodule of .

Proposition: The following conditions on a left -module are equivalent:

1. is semisimple.
2. is a direct sum of simple modules.
3. is generated by its simple submodules.

Proof. : Let be the sum of the simple submodules of . By assumption there is a direct sum decomposition . Since is semisimple, it is either zero or has a simple submodule, but by assumption the latter is not possible, so .

: Let be generated by its simple submodules . Let be a submodule of . By Zorn’s lemma, there is a maximal collection of simple submodules such that is an internal direct sum and such that . Let

.

If , then there is some not contained in , hence , which contradicts the maximality of . So as desired.

: apply the above argument to .

: obvious.

Semisimple rings

Ideally, we’d like to be able to study all the modules of a ring by studying its semisimple modules. This desire is encapsulated by the following definition.

Theorem-Definition: A (left) semisimple ring is a ring satisfying any of the following conditions, all of which are equivalent:

1. All left -modules are semisimple.
2. All finitely-generated left -modules are semisimple.
3. All cyclic left -modules are semisimple.
4. is semisimple as a left -module.

Proof. There are obvious implications . Since we know that semisimplicity is preserved under taking quotients, we obtain . Since any module is generated by the modules where is a set of generators, we obtain .

By Maschke’s theorem, the group algebra is semisimple for a finite group and a field of characteristic not dividing , so it is clearly of interest to understand the structure of semisimple rings.

Example. Any division ring is simple, hence semisimple, as a module over itself, so any division ring is semisimple.

Example. Let be two semisimple rings. In the product ring , consider the two idempotents . Any -module breaks up into a direct sum where the first factor is the part on which acts nontrivially and the second is the part on which acts nontrivially. In particular, is semisimple as a module over itself, hence is semisimple, and moreover its simple modules are precisely the simple modules of together with the simple modules of .

Example. Let denote the matrix ring of matrices over a division ring . As a vector space it is spanned by elements which satisfy the relations

By inspection, as a left module over itself, breaks up into a direct sum of simple submodules . It follows that is semisimple. Moreover, since every simple module is a quotient of (as a left module over itself), it follows that all simple -modules are isomorphic to some , each of which is in turn isomorphic to , which admits a natural left action of when thought of as a right -module.

Artin-Wedderburn

The above discussion shows that any finite product of matrix rings over division rings is semisimple. In fact, this exhausts all examples. To see this, we first need the following.

Cayley’s theorem for rings: .

Proof. acts by right multiplication on , and each such right multiplication is a left -module homomorphism, so certainly contains . On the other hand, if is a left -module homomorphism, then is already given by right multiplication.

Theorem (Artin-Wedderburn): Every semisimple ring is isomorphic to a finite product of matrix rings over division rings. Moreover, the terms in this product are uniquely determined up to permutation.

Proof. Let be semisimple. Then , as a left -module, admits a direct sum decomposition into simple submodules. The multiplicative unit of generates it as a module, hence generates any quotient of it, so has a nonzero image in any simple quotient of ; it follows that the direct sum decomposition of into simple submodules is finite, so we can write

for some positive integers and some simple modules . Then

by Schur’s lemma and the universal property of direct sums. The opposite of a matrix ring over a division ring is a matrix ring over the opposite division ring, so we conclude that is isomorphic to a finite product of matrix rings over division rings. Moreover, by the above remarks, if

is a finite product of matrix rings over division rings, then the simple modules over are determined by the simple modules over each factor. The only simple module over , as we have seen, is , and each simple module is cyclic hence appears as a direct summand of , so the product decomposition above is unique up to permutation by Jordan-Hölder.

As a corollary, a ring is left semisimple if and only if it is right semisimple, so we can drop the adjective.

satisfying the identities

1. ,
2. ,
3. ?

A moment’s reflection shows that there isn’t such a group; the existence of the groups , where is an arbitrary set, shows that there exist groups of arbitrarily large cardinality, so no particular group can be universal.

However, it still seems like we can say things about the universal group, even though it doesn’t exist. That is, given a first-order statement in the language of groups, we can still give a sensible definition of what it means for that statement to be true in the universal group: it merely has to be true of all groups! Equivalently, by the completeness theorem for first-order logic, it merely has to be deducible from the group theory axioms. For example, it’s true in any group that

so this statement must be true in the universal group.

In fact, the universal group does exist: it’s just not a set! Namely, we can write down all of the objects together with all of the maps implied to exist by the existence of the structure maps , such as the map

,

and together with all of the equalities between these maps implied by the group theory axioms. This data describes a category , in fact a monoidal category, that can be succinctly described as the free monoidal category on a group object. The object is a group object which deserves the name “universal group” in the sense that, given any other monoidal category , the category of monoidal functors and monoidal natural transformations is equivalent to the category of group objects in .

The category is the Lawvere theory describing the theory of groups. It can be described more concretely as the opposite of the category whose objects are the free groups and whose morphisms are all group homomorphisms between these; indeed, the components of the universal maps are just -tuples of elements of , which are the same thing as homomorphisms , and it’s not hard to see that this identification respects composition.

Describing groups in this way has, to my mind, one major conceptual benefit over the standard definition: it emphasizes that the choice to present the theory of groups using a particular set of maps and axioms is just as arbitrary as the choice to present a group using a particular set of generators and relations. As long as one gets the same Lawvere theory in the end, one is still studying groups. For example, instead of using identity, multiplication, and inversion, we can use identity and the ternary map together with the heap axioms.

.

The intermediate value theorem shows that , which was the subject of a recent math.SE question that provided the inspiration for this question. I provide a stronger lower bound on using elementary inequalities and entirely by hand in an answer to the linked question, although I don’t try to improve the upper bound.

My current plans involve going through my backlog of books and papers I haven’t had time to read and writing posts about them, but I’m sure there are plenty of other ways I could mathematically enrich my life before graduate school and I’d be very interested to hear your suggestions.

I’m glad I finally have a word for this. I’ve cared about moral truth more than proof for awhile now, and that’s a major reason I’ve been trying to teach myself physics: even if it isn’t a good source of proofs, it seems like a great source of moral truths.

More generally, if are Poisson algebras, then the tensor product can be given a Poisson bracket given by extending

linearly. At least when are unital, this Poisson algebra is the universal Poisson algebra with Poisson maps from such that the images of elements of Poisson-commute with the images of elements of . In particular, it follows that is a Poisson algebra with the bracket

.

This describes a classical particle in dimensions, or different classical particles in one dimension, and it is the classical limit of a quantum particle in dimensions, or different quantum particles in one dimension.

Today we’ll discuss the question of how one might go about constructing Poisson brackets more generally.

Alternating biderivations

Recall that an alternating biderivation on an algebra is an alternating bilinear map which is a derivation in each variable. Furthermore, recall that if are derivations on , then is an alternating biderivation. Thus we get a natural alternating map which factors through the exterior square

.

This map is injective since for all if and only if are scalar multiples of each other. A natural question is when this map is also surjective.

Proposition: The above map is surjective when is a polynomial algebra.

Proof. Any alternating biderivation on is determined by . Furthermore,

is an alternating biderivation such that and such that for all . It follows that for any choice of elements there exists a unique alternating biderivation satisfying given by

.

This is a very special case of the Hochschild-Kostant-Rosenberg theorem, at least once you know that the Harrison cohomology of a polynomial algebra vanishes. But this isn’t difficult to see: any commutative first-order deformation is necessarily itself a polynomial algebra because no relations can exist between any lifts of the generators.

The following alternate perspective on the above result may be enlightening. Note that any derivation factors through the -module generated by formal symbols of the form subject to the following relations:

1. whenever .
2. .
3. .

This -module is denoted and known as the space of Kähler differentials. Since all of the above axioms are satisfied by the exterior derivative of a function on a smooth manifold, the intuition here is that is analogous to the space of differential -forms on . This fits in nicely with the intuition that derivations are analogous to vector fields on , since by definition a derivation is an -module morphism , so there is a natural pairing between the two.

But now it’s clear that an alternating biderivation on is nothing more than an -module morphism

.

Thus alternating biderivations are dual to -forms. Intuitively, they are therefore bivector fields on . If , as an -module, behaves sufficiently similar to a finite-dimensional vector space over a field, then it follows that the dual of its exterior square ought to be isomorphic to the exterior square of its dual, which is what we showed above when is a polynomial algebra. In this case, is in fact a free -module on generators , which is what makes the above argument work abstractly. More generally I think the above argument goes through whenever is finitely-generated and projective.

(Above I am glossing over the distinction between the exterior square and the space of alternating -tensors. I haven’t yet made up my mind about when this distinction is worth making.)

The Jacobi identity

Now that we have a reasonably good grasp of alternating biderivations, what can we say about the ones that satisfy the Jacobi identity (and therefore are Poisson brackets)?

Proposition: An alternating biderivation on an algebra satisfies the Jacobi identity if and only if the Jacobi identity is satisfied when a set of generators of is plugged in.

Proof. It suffices to observe that the Jacobiator

is a triderivation: it is trilinear and satisfies the Leibniz rule in each of its components separately. (This is a straightforward computation using the Leibniz rule for .) Thus it is determined linearly by its values on a set of generators of .

We record the following two immediate corollaries.

First, if is a vector space equipped with an alternating bilinear form , then extends to a Poisson bracket on the symmetric algebra . (The Jacobi identity is clearly satisfied on generators since is a scalar for any .) These are precisely the polynomial Poisson algebras for which the Poisson bracket is graded with degree . In particular, if has a basis such that is given by

then we get precisely the algebra of observables on classical particles as described earlier.

Second, if is a Lie algebra, then the Lie bracket extends to a Poisson bracket on the symmetric algebra . These are precisely the polynomial Poisson algebras for which the Poisson bracket is graded with degree .

In both of these cases we can explicitly find a deformation quantization: that is, we can identify a formal deformation from which we get the above Poisson algebras as classical limits. This will be expanded on in later posts.

However, if is commutative, all commutators are trivial, and yet classical mechanics somehow takes a Hamiltonian and produces a notion of time evolution. How does that work? It turns out that for algebras of observables of a classical system, we can think of as the classical limit of a family of noncommutative algebras. While is commutative, the noncommutativity of the family equips with the extra structure of a Poisson bracket, and it is this Poisson bracket which allows us to describe time evolution.

Today we’ll describe one way to formalize the notion of taking the classical limit using the deformation theory of algebras. We’ll see how Poisson brackets pop out along the way, as well as the relevance of the lower Hochschild cohomology groups.

Hochschild cohomology

In this post we consider associative, but not necessarily unital, algebras over a fixed field of characteristic not equal to .

Let be such an algebra. A formal deformation of is an associative -bilinear product on (when is noncommutative, should be central) such that for we have

.

In other words, quotienting by we obtain the original product on . For the purposes of understanding the classical limit, should be thought of as the classical algebra of observables and , together with the deformed product , should be thought of as the quantum one. Roughly speaking, formal deformations of an algebra describe the “formal neighborhood” of in the moduli space of algebras.

Formal deformations turn out to be difficult to construct in general. As a first approximation, instead of quotienting by we can quotient by . A first-order deformation of is an associative -bilinear product on satisfying the same condition as above. Explicitly, it is given by a product

where is a -bilinear map such that , or equivalently

.

The -vector space of all such maps is denoted and called the space of Hochschild -cocycles of with coefficients in . Roughly speaking, first-order deformations of describe the “tangent space” to in the moduli space of algebras.

Of course, this is not really the space we’re interested in. If we actually want to understand first-order deformations of , then it would be a good idea to identify -cocycles that give isomorphic deformations. Hence suppose are -cocycles describing deformations which are isomorphic in the sense that there is a -linear map sending one to the other which reduces to the identity . Writing , we want

which after simplification gives

.

In other words, give isomorphic deformations if and only if lies in the subspace of -cocycles of the form for some -linear map . These -cocycles are precisely the -coboundaries ; as -cocycles, they describe the deformations isomorphic to the trivial deformation given by with its usual product. The quotient space , the second Hochschild cohomology , is the desired space which parameterizes first-order deformations of .

It’s worth mentioning at this point that Hochschild cohomology can be computed from a cochain complex beginning

where sends an element to a map and sends a map to a map (at least up to a sign I may have wrong). The kernel of is the space of Hochschild -cocycles, and its quotient by the image of gives as above.

But there are two smaller cohomology groups we can describe reasonably concretely as well. The kernel of defines zeroth Hochschild cohomology

which is precisely the center of . The kernel of defines the space of Hochschild -cocycles , which is precisely the space of derivations . The image of defines the space of Hochschild -coboundaries , which are given by inner derivations (named by analogy with inner automorphisms). The quotient defines the first Hochschild cohomology

which is precisely the space of outer derivations of .

Introducing the lower Hochschild cohomology groups gives Hochschild cohomology extra structure. For reasons I don’t particularly understand, if are two derivations, then

is a Hochschild -cocycle. Moreover, if either of the derivations is inner, then turns out to be a -coboundary. So the above map descends to a “cup product”

.

Hodge decomposition

Now suppose is commutative. Taking the opposite algebra of a first-order deformation gives another first-order deformation; concretely, if is a -cocycle, then is also a -cocycle. Then (and this is where the hypothesis that is important) every -cocycle admits a canonical decomposition

into symmetric and alternating -cocycles. Moreover, since is commutative, trivial deformations of are commutative, so the -coboundaries all lie in the symmetric subspace. Hence we can identify a direct summand of the second Hochschild cohomology consisting of the quotient of the symmetric -cocycles by the -coboundaries. This is known as second Harrison cohomology, and describes commutative deformations of .

On the other hand, since all -coboundaries are symmetric, it follows that the alternating -cocycles map injectively into Hochschild cohomology. Moreover, since

is a commutator, it follows that alternating -cocycles are biderivations (derivations in each variable separately). On the other hand, it is straightforward to verify that any alternating biderivation is a -cocycle. It follows that we have a “Hodge decomposition”

.

Notably, is itself a space of functions rather than a quotient of a space of functions by another space of functions, so it is a little more concrete to work with than Hochschild or Harrison cohomology. In addition, biderivations are uniquely determined by what they do to generators of , so it is easier to write down biderivations than general -cocycles.

Example. If are two derivations , then is an alternating biderivation. Note that this is just the image of the cup product in .

As it turns out, a large class of commutative algebras, the “smooth algebras,” have the property that (so have no nontrivial commutative first-order deformations). For such algebras, can be understood very concretely as the space of alternating biderivations. I believe such algebras include the algebras of functions on a smooth affine variety, but I don’t understand these issues well yet.

Poisson algebras

A second-order deformation of is an associative -bilinear product on such that quotienting by we obtain the original product on . Writing such a deformation as

its first-order part is necessarily a Hochschild -cocycle, but one which satisfies an additional condition: the second-order part of the identity is gives

.

Functions of the form on the RHS are Hochschild -coboundaries, and in fact this condition can be interpreted to mean that the associator of must be zero in (which it would take us too far afield to define here). In particular, if , then every first-order deformation of extends to a second-order deformation.

The above condition on implies another condition which we are more interested in for the time being. Namely, the commutator

has first-order part , and after a quick calculation, it turns out that the fact that satisfies the Jacobi identity implies that does as well.

This suggests the following definition. A Poisson bracket on is a Lie bracket which is also a biderivation. In formulas, is a -bilinear map satisfying

1. 
2. 
3. .

A Poisson algebra is an algebra equipped with a Poisson bracket. Any noncommutative algebra has a canonical Poisson bracket given by the commutator , but significantly, even if is commutative, it can still admit a nontrivial Poisson bracket if it admits any second-order deformations which are noncommutative to first order.

Poisson algebras are the natural setting for the most general form of the Heisenberg picture. Given a Poisson algebra and a Hamiltonian , the derivation is now the infinitesimal generator of time evolution. The procedure above, where we took the first-order part of a noncommutative deformation, also elegantly explains how Poisson brackets on a quantum system descend to Poisson brackets on a classical system.

Hamiltonian mechanics

Let’s see how this works more explicitly. We return to the setting of a quantum particle in one dimension in a potential. Recall that in this case the Hamiltonian is given by

where are multiplication operators and . Suppose for simplicity that is a polynomial in . Then all of the operators we care about are more or less contained in the algebra of operators generated by subject to the relation

.

Taking the classical limit as above with we obtain the commutative algebra with Poisson bracket uniquely defined by and extended via the Leibniz rule. Recall that is a Poisson bracket for any two derivations ; this particular Poisson bracket can be written

for all . We find that

.

Since we divided by , the infinitesimal generator of time evolution is now given by , and so we have recovered Hamilton’s equations (as well as Newton’s second law, again) from the Heisenberg picture.

Note that the Poisson bracket defined above naturally extends to the algebra of smooth functions on the classical phase space of a classical particle in one dimension. More generally, given a smooth manifold , there is a natural Poisson algebra structure on the algebra of smooth functions on the cotangent bundle of induced by a canonical symplectic form. In other words, the cotangent bundle is a Poisson manifold.

A natural question to ask about Poisson manifolds, and more generally about commutative Poisson algebras, is whether they admit a unique formal deformation whose first-order part gives the corresponding Poisson bracket; this is then the unique deformation quantization of the classical system described by the Poisson algebra. That this is true for Poisson manifolds is a deep result of Kontsevich.

Curiously enough, the Zipf distribution which shows up in that post is the same as the zeta distribution that shows up when trying to motivate the definition of the Riemann zeta function. I’m sure there is a conceptual explanation of this connection somewhere, probably coming from statistical mechanics, but I don’t know it. I suppose the approximate scale invariance of the zeta distribution is relevant to its appearance in many real-life statistics, as described in Terence Tao’s blog post on the subject here.

where is a self-adjoint operator on called the Hamiltonian. Observables are given by other self-adjoint operators , and at least in the case when has discrete spectrum measurement can be described as follows: if is a unit eigenvector of with eigenvalue , then takes the value upon measurement with probability ; moreover, the state vector is projected onto .

The Heisenberg picture is an alternate way of understanding time evolution which de-emphasizes the role of the state vector. Instead of transforming the state vector, we transform observables, and this point of view allows us to talk about time evolution (independent of measurement) without mentioning state vectors at all: we can work entirely with the algebra of bounded operators. This point of view is attractive because, among other things, once we isolate what properties we need this algebra to have we can abstract them to a more general setting such as that of von Neumann algebras.

In order to get a feel for the kind of observables people actually care about, we won’t study a finite toy model in this post: instead we’ll work through some classical (!) one-dimensional examples.

The Heisenberg picture

From our above description of measurement of an observable it follows that the expected value can be given by the elegant formula

.

Since the characteristic function of a random variable completely determines it, it is possible in principle to replace knowledge of the state vector with knowledge of the expectation it induces on observables. In the theory of von Neumann algebras, is referred to as a state for this reason.

Now, as evolves according to the Schrödinger equation, the expectation of an observable evolves as

.

In the Schrödinger picture, we keep our algebra of observables invariant and modify the state (or equivalently the expectation ), but in the Heisenberg picture, we keep the expectation invariant and modify the algebra of observables by the one-parameter group of inner automorphisms

.

This is quite an elegant way to think about time evolution: it tells us that we can delay thinking about states until we actually want to compute probabilities. At any point before then, we can think directly about how observables are changing, and consequently we don’t need to mention states at all to talk about properties of observables which don’t depend on initial conditions.

Now that we’ve conceived of time evolution as a one-parameter group of inner automorphisms of the algebra of observables, we can take its derivative, which is precisely the inner derivation

by a simple calculation. Recalling that, for a one-parameter group of automorphisms with associated derivation , we have at least formally the Taylor expansion

it follows that

.

In particular, time evolution is completely determined by the commutator of the Hamiltonian with any observable. Note that is invariant under time evolution if and only if it commutes with the Hamiltonian, and consequently note that the infinitesimal generator of any one-parameter group of symmetries of the Hamiltonian gives a self-adjoint operator by Stone’s theorem, hence gives a conserved observable. This is how Noether’s theorem appears in quantum mechanics.

Particle on a line

The most basic example is the free particle on . Here the Hamiltonian is where is mass (a constant) and is momentum (an observable), which is, up to normalization, the infinitesimal generator of translation. (Note that translation clearly acts by a one-parameter group of unitary transformations on , so Stone’s theorem assures us we have a self-adjoint operator here.) This gives

which is, up to normalization, the ordinary Laplacian. Formally, the eigenvectors of with non-negative eigenvalues are with corresponding eigenvalues

.

We have chosen these eigenvectors because they are also eigenvectors of the momentum operator with eigenvalues , which recovers the de Broglie relation . (The other de Broglie relation, , is already built into the Schrödinger equation since an eigenvector for the Hamiltonian with eigenvalue is multiplied by in time evolution.) Note that these “eigenvectors” do not actually exist in . There is a formalism for dealing with this, but I don’t know it; in any case it can be dealt with.

Time evolution for a single eigenvector is given by

.

Of course multiplying the state vector by a nonzero complex number doesn’t affect anything we can measure about the state. It’s only when two or more eigenvectors are added together that the multiplication above manifests itself as (what appears to be) interference between waves. In fact, the above describes precisely a plane wave with angular frequency and wavenumber .

Since is a scalar multiple of the square of , it follows that , so we recover conservation of momentum. The operator (more precisely, multiplication by ), whose eigenvalues describe the position of our free particle, should not commute with either or since it should not be conserved. Instead, we have the canonical commutation relation

which gives (using the fact that is a derivation)

.

This gives

so we recover the familiar fact that is the velocity of a free particle.

Particle in a box

In order to work with some eigenvectors which actually exist in (hence which give rise to well-defined probability distributions), let’s restrict our formerly free particle to a box . This is equivalent to requiring that the state vector vanish outside of this interval. If we further require that the state vector is continuous, then it must vanish at and . The eigenspace of the Hamiltonian with eigenvalue is spanned by , and in each of these eigenspaces we can find normalized eigenvectors

vanishing at and whenever . This is a complete list of eigenvectors of the Laplacian vanishing at and , and time evolution is given by

where

.

Now that we’ve restricted ourselves to a compact space, we finally get a system in which the energy eigenvalues are discrete, or quantized (from which quantum mechanics gets its name). Intuitively speaking, a particle in a box behaves like a wave, constantly bouncing against the walls; if it oscillates at the wrong frequency, destructive interference would eventually cause it to disappear. Only certain frequencies, dictated by the shape of the box, survive.

Note that the lowest energy eigenvalue is not zero; it occurs when . So unlike the classical case, a quantum particle in a box cannot have zero energy.

Particle in a potential

Next we consider the case when a particle on is subject to some potential. Potentials modify momentum, but the relationship between momentum and velocity should remain intact, so we still want

hence we still want . We can guarantee this whenever has the form

where is a multiplication operator, since , and in fact is the desired potential. Potentials, being not translation-invariant in general, break conservation of momentum, and we instead have . Now, let us suppose that

is a nice analytic function of . By induction on the relation , using the fact that is a derivation, we conclude that , hence (at least formally)

.

From this it follows that

which recovers Newton’s second law , remembering that classically.

We can deduce all this despite the fact that for arbitrary it is not at all obvious how to directly write down the eigenvectors or eigenvalues of the corresponding Hamiltonian.

A particularly simple case occurs when for some constant . Then we compute that , hence that

exactly as in the classical case. I do not know what the eigenvectors of the Hamiltonian look like in this case.

Another simple case of a particle in a potential is the quantum harmonic oscillator, where for some constant , so that

.

This gives

which, together with the relation , implies that the iterated commutators of either or with are (up to normalizing factors) periodic, alternating between some constant times and some constant times . In fact, we have

again exactly as in the classical case of, for example, a spring-mass system. Note that in this special case, conservation of energy is equivalent to the Pythagorean theorem!

The quantum harmonic oscillator is important in quantum field theory for reasons I don’t understand yet. Part of its basic importance to quantum mechanics can be understood as follows: for an arbitrary potential, expanding out in the neighborhood of a critical point, the linear term vanishes and so generically we get a quadratic term (plus lower order terms), hence to second order a harmonic oscillator.

There is a very elegant way to write down the eigenvectors of the Hamiltonian in this case using Dirac’s ladder method, but I think it would be best to leave such considerations until I understand the harmonic oscillator better.

Generalizations

One can take tensor products of any of the examples above to get examples in higher dimensions; for example, one can can consider particles in boxes in for any . More generally one can consider particles on any Riemannian manifold, where the kinetic term of the Hamiltonian is taken to be a suitable multiple of the Laplacian. Particularly interesting cases include manifolds with large symmetry groups, since the eigenspaces of the Laplacian break up into irreducible representations of these groups.

At the end of the post we’ll briefly describe what we can conclude from all this about electrons orbiting a hydrogen atom.

Generalities

Below, “representation” means “finite-dimensional complex continuous representation.”

Let be a compact group and suppose that, one way or another, we have found a (left-invariant) Haar measure on , normalized to have total measure . Such measures exist for all compact groups but are non-trivial to construct in general; in the case of the Haar measure can be straightforwardly described as the measure on inherited from Lebesgue measure on divided by the volume of :

.

It follows that given a representation of , we can define the averaging operator

.

As in the case of finite groups, the averaging operator is a projection from onto its invariant subspace. It follows that we can average an inner product on so that it is -invariant, hence WLOG we talk about unitary representations only and Maschke’s theorem holds: representations are completely reducible. Schur’s theorem holds in this setting with exactly the same proof as usual.

Also as for finite groups, taking the trace of a representation defines its character , a continuous class function . Taking characters is additive under direct sum, multiplicative under tensor product, and conjugate under taking duals. Moreover, given two representations on vector spaces we can define the inner hom (what I’ve denoted by something like in previous posts, but I think this notation is less confusing), which explicitly is the space of linear transformations with action given by

and which abstractly is determined by the adjunction

.

In particular, , so the invariant subspace of can be identified with the space of -morphisms . On the other hand, is isomorphic to , hence if denote the corresponding characters, has character . Since the trace of the averaging operator gives the dimension of its image (the invariant subspace of a representation), it follows that

and combining this result with Schur’s lemma, the orthogonality relations follow exactly as for finite groups. In particular, a representation of is uniquely determined up to isomorphism by its character.

The irreducible representations of

has an obvious -dimensional irreducible representation which we’ll denote by . To specify the character of any representation of , it suffices to specify its restriction to the maximal torus of elements of the form since every element is conjugate to an element of the torus, and the character of is then .

From we can construct some additional representations: we have where the former has dimension and the latter has dimension . It is not hard to see that the abelianization of is trivial, so is trivial, hence is self-dual and quaternionic. It follows that cannot have a one-dimensional summand, hence is an irreducible representation of dimension . If has eigenvalues , then its action on has eigenvalues , so the character of this representation is given by

.

Note that the double cover gives an irreducible -dimensional real representation of which extends to a -dimensional complex representation. In this representation acts by rotation by , where , and it follows that the character of this representation agrees with , hence that the two are isomorphic.

What can we say about higher-dimensional representations? Well, given , a standard strategy for constructing more representations is to apply various functors to . Here it will be productive to consider the symmetric powers , getting the hint from . Since , these representations have dimension . Note that is the trivial representation by definition. Recall that if is a diagonalizable linear operator with eigenvalues , then has eigenvalues the products of all unordered -tuples of eigenvalues of (and this is easily proven by exhibiting the corresponding eigenvectors). It follows that the characters of the representations are given by

.

If , the above is therefore equal to where is the Chebyshev polynomial of the second kind.

Proposition: All of the representations are irreducible.

Proof. Since has real character, it is self-dual, so we can identify with the space of homogeneous polynomials of degree on . If is equipped with an invariant inner product, then letting denotes an orthonormal basis for , acts on the symmetric algebra

of polynomial functions on as follows:

.

(The matrix given is the matrix of an element of with respect to an orthonormal basis such that is the corresponding dual basis.) Let be a nonzero homogeneous polynomial of degree . A minimal invariant subspace containing must also contain for every , and by taking appropriate linear combinations of these polynomials, contains every monomial in . Thus we may assume WLOG that is a single monomial . But for sufficiently small the action of sends to a polynomial all of whose coefficients are nonzero, hence contains every monomial in , so as desired.

It follows that the characters are orthogonal and satisfy . This can also be checked with an explicit formula for integrating a class function on , but we will not need to do this.

The representations all have real characters, so are all self-dual. A computation of the characters of both sides, or an explicit argument with bases, shows that

.

It follows by induction that any irreducible subrepresentation of is isomorphic to for some . Since they are all self-dual, these are in some sense all of the representations of obtainable from via universal methods.

Theorem: Every irreducible representation of is isomorphic to for some .

Proof 1

Our first proof is based on the following important observation: since the character of a representation of is determined by its restriction to any maximal torus , the restriction functor is essentially injective. So we should try to understand the second category in order to understand what kind of characters are possible.

Theorem: Regard as the unit complex numbers . Then every irreducible representation of is isomorphic to the representation for some .

This result is closely related to the existence of Fourier series; see Pontryagin duality for a general discussion.

Proof. Since is abelian it is true as for finite groups that any irreducible representation has dimension (since any eigenvector of a non-identity element spans an invariant subspace). Thus an irreducible representation is just a continuous homomorphism . By compactness the image of this map is contained in , so an irreducible representation is just a continuous homomorphism . By path lifting, lifts to a continuous homomorphism , which must therefore be of the form for some real . This gives

which is a homomorphism if and only if , and the conclusion follows.

Any irreducible representation of decomposes into the direct sum of finitely many irreducible representations of a given maximal torus . Suppose the representation occurs times. Then the character of must be given by

where has eigenvalues . That is, it must be a Laurent polynomial in with integer coefficients. Moreover, since we can swap the order of the eigenvalues, it follows that is invariant under the substitution , so it must be a symmetric Laurent polynomial in :

.

However, it is not hard to see that the characters of the symmetric powers give a -basis of the symmetric Laurent polynomials in , so it follows that there exist such that

.

In particular, for some , hence for some as desired.

In general, maximal tori are very important in the classification and representation theory of compact Lie groups. The Lie-algebraic analogue of a maximal torus is a Cartan subalgebra, which plays a corresponding role in the classification and representation theory of semisimple Lie algebras.

Proof 2

Our second and third proofs are both motivated by the standard result that if is a finite group and a faithful representation of , then every irreducible representation of occurs in for some . This result can be proven in many ways, some of which don’t generalize to compact groups, and in fact the result is not true as stated for compact groups: the example of and the representation shows that we need to at least consider for integers . This result is true (see for example this MO question), but Proof 2 will not be enough to prove it in general. Nevertheless, it works in the special case of .

First, note that if and only if they have the same eigenvalues (which are determined by their real part), hence if and only if they are conjugate in . So letting denote the eigenvalues of , the space of conjugacy classes of is naturally identified with , and separates points on this space. It follows by the complex Stone-Weierstrass theorem that the smallest subalgebra of the algebra of class functions containing and closed under conjugation is dense in the space of continuous class functions with the uniform norm. But since is self-dual, this algebra is spanned by the characters of the representations , which are all direct sums of the representations .

It follows that given an irreducible representation , we can find a sequence of class functions which are linear combinations of the characters converging uniformly to . It again follows that for some , and again we are done.

This proof is not enough to give the general result about faithful representations because does not necessarily separate conjugacy classes in general. However, it is still possible to use the Stone-Weierstrass theorem to prove the general result, and this is the subject of the next proof.

Proof 3

Our third proof relies on a companion result to the orthogonality relations for characters. For us, a matrix coefficient of a compact group is a continuous function of the form for a unitary representation and vectors .

Theorem (orthogonality for matrix coefficients): Let be matrix coefficients of a compact Lie group , where are irreducible unitary representations on with characters . If are non-isomorphic, then

.

(The full statement of the orthogonality relations includes an expression for the inner product of matrix coefficients from the same irreducible representation, but there is a constant in it that I can’t figure out, and in any case we won’t need it.)

Proof. Our convention is that inner products are conjugate-linear in the first variable. Fix and . We can write

.

For fixed , the integrand gives a bilinear map (since it is linear in but conjugate-linear in ); moreover, the natural action of by precomposition gives a representation isomorphic to on the space of such bilinear maps, and the above integral computes precisely the averaging operator on this representation. If is not isomorphic to , the corresponding tensor product does not have any copies of the irreducible representation, so the averaging operator sends any vector to zero and the conclusion follows.

Theorem: Let be a compact group with a faithful representation . Then the span of the matrix coefficients of the representations is dense in .

Proof. The sum of any two matrix coefficients of a given representation is another matrix coefficient of the same representation . Furthermore, the complex conjugate of a matrix coefficient of is a matrix coefficient of , and the product of a matrix coefficient of and a matrix coefficient of is a matrix coefficient of . It follows that if has a faithful representation , then by the complex Stone-Weierstrass theorem, the space spanned by the matrix coefficients of the representations is dense in . In particular, if is any other irreducible representation, then we can approximate its matrix coefficients with the above matrix coefficients, and we find by orthogonality that is a subrepresentation of for some .

Proof 4

For our final proof we will provisionally assume that all representations are smooth so that we can pass from a representation to the induced map on Lie algebras . As it turns out, all continuous homomorphisms between Lie groups are automatically smooth (although we won’t prove this), so we can assume this WLOG.

Recall that a representation of a Lie algebra is a homomorphism of Lie algebras for a (finite-dimensional complex) vector space. There is an obvious notion of direct sum of representations. As for groups, we say that a representation is irreducible if it has no non-trivial -invariant subspaces.

Proposition: Let be a Lie group all of whose elements are contained in a one-parameter subgroup of , a smooth representation, and the induced representation of . Then is irreducible as a representation of if and only if it is irreducible as a representation of .

Proof. By assumption, is irreducible if and only if it has no non-trivial subspaces invariant under all of the one-parameter subgroups of . Let be such a one-parameter subgroup. Let be a nonzero vector. Then

and

.

It follows that any point in the minimal -invariant subspace containing is a limit of points in the minimal -invariant subspace containing , and vice versa, hence that the two coincide.

In particular, we now know that the representations of associated to are all irreducible. Moreover, given a representation of which is known to come from a representation of , by exponentiating we can recover the action of a maximal torus, hence the character of the representation. It follows that if we show that every irreducible representation of is of the form , we have also shown the same statement for . (Note that we don’t need to know how to lift representations of to representations of to do this.)

Recall that can be explicitly presented as the imaginary quaternions under the inherited bracket

.

(I’m not using the symbols because we’re about to extend scalars to .) Since we only care about complex representations of this real Lie algebra, we can pass to the complexification

(a technique which highlights the algebraic convenience of working with Lie algebras), and now that we are working over we can ask for a more convenient basis of . One idea is to look at the adjoint action . By inspection the eigenvectors of this linear transformation are with eigenvalues respectively. So one convenient choice of basis is , in which the relations are given by

.

This basis is convenient for the following fundamental reason. Let be any representation of . (We won’t distinguish between an element of and its action on .) Suppose is an eigenvector of of eigenvalue . Then

and similarly

hence is (zero or) an eigenvector of of eigenvalue and is (zero or) an eigenvector of of eigenvalue ! The eigenspaces spanned by these eigenvectors are known as the weight spaces of , and correspond to the decomposition of under the action of a maximal torus of . But unlike in the first proof above using maximal tori, we can now use elements of the Lie algebra to move between weight spaces.

The sequence of vectors are all eigenvectors with distinct eigenvalues, so since is finite-dimensional the sequence must eventually terminate. Hence we may assume WLOG that . In this case is known as a highest weight vector. Since , it follows that

is a -invariant subspace of . If is irreducible it must therefore be all of . If , we conclude that

.

Hence has a basis of eigenvectors of with eigenvalues . In particular, it follows that the trace of acting on is

.

But since and the trace of a commutator is zero, it follows that we must have . The resulting weights are precisely the weights of , and it follows that as desired.

Note that, unlike all of the other proofs, this one explicitly constructs the irreducible representations (at least as representations of the Lie algebra) without requiring that we knew what they were in advance.

The representation theory of

An electron orbiting a hydrogen atom can be described by four quantum numbers, which are really labels for a direct sum decomposition of the corresponding Hilbert space of states. Using what we now know about the representation theory of , we can explain two of these quantum numbers, although we will do so in more detail in a later post.

Keeping in mind the double cover , any irreducible representation of pulls back to an irreducible representation of . The ones we get in this way are precisely the ones in which acts trivially, and by inspection these are the irreducible representations . Hence has exactly one irreducible representation of each odd dimension . (Of course all of the irreducible representations of are projective representations of , so we should expect the even-dimensional ones to also be important in quantum mechanics.)

It follows that in any quantum mechanical system with physical symmetry, the eigenstates corresponding to a particular energy eigenvalue will organize themselves into clumps with an odd number of members. For electrons around a hydrogen atom, the corresponding clumps are referred to in chemistry as subshells and delineated by their azimuthal quantum number , which measures orbital angular momentum. In chemistry, subshells are also traditionally referred to by letters based on what the corresponding atomic spectra looked like:

1. (sharp) refers to ,
2. (principal) refers to ,
3. (diffuse) refers to ,
4. (fundamental) refers to ,

and so forth. Experimentally, subshells hold electrons, subshells hold electrons, subshells hold electrons, subshells hold electrons, and so forth. This is off by exactly a factor of from what can be predicted purely on the basis of symmetry (where we would expect electrons, respectively), and this is because there is a second symmetry here coming from spin. Nevertheless, the representation theory of alone is already enough to address abstractly the origin of one quantum number, and in fact by fixing a maximal torus and considering its eigenvectors we get another one, the magnetic quantum number . The reason it takes values between and is that these are the possible weights of .

Since an argument based on symmetry is independent of the choice of Hamiltonian, we cannot use symmetry alone to figure out what the energy eigenspaces are, so we still can’t explain the principal quantum number without actually looking at the Hamiltonian, and we also haven’t explained spin. But two out of four isn’t bad!