Quantum Mechanics for Mathematicians: The Heisenberg group and the Schrödinger Representation

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1 Quantum Mechanics for Mathematicians: The Heisenberg group and the Schrödinger Representation Peter Woit Department of Mathematics, Columbia University November 30, 2012 In our discussion of the free particle, we used just the actions of the groups R 3 of spatial translation and the group R of time translation, finding corresponding observables, the self-adjoint momentum (P) and Hamiltonian (H) operators. We ve seen though that the Fourier transform involves a perfectly symmetrical treatment of position and momentum variables. This allows us to introduce a position operator Q acting on our state space H. We will analyze in detail in this chapter the implications of extending the algebra of observable operators in this way, most of the time restricting to the case of a single spatial dimension, since the physical case of three dimensions is an easy generalization. The P and Q operators generate an algebra named the Heisenberg algebra, since Werner Heisenberg and collaborators used it in the earliest work on a full quantum-mechanical formalism in It was quickly recognized by Hermann Weyl that this algebra comes from a Lie algebra representation, with a corresponding group (called the Heisenberg group by mathematicians, the Weyl group by physicists). The state space of a quantum particle, either free or moving in a potential, will be a unitary representation of this group, with the group of spatial translations a subgroup. Note that this particular use of a group and its representation theory in quantum mechanics is both at the core of the standard axioms and much more general than the usual characterization of the significance of groups as symmetry groups. While the group of spatial translations is a symmetry of the theory of a free particle, the action of the larger Heisenberg group on the state space H does not commute with any non-zero Hamiltonian operator. The Heisenberg group does not in any sense correspond to a group of invariances of the physical situation, but rather plays a much deeper role. 1

2 1 The position operator and the Heisenberg Lie algebra In the description of the state space H as functions of a position variable q, the momentum operator is P = i d dq The Fourier transform F provides a unitary transformation to a description of H as functions of a momentum variable p in which the momentum operator P is just multiplication by p. Exchanging the role of p and q, one gets a position operator Q that acts as Q = i d dp when states are functions of p (the sign difference comes from the sign change in F vs. F), or as multiplication by q when states are functions of q. 1.1 Position space representation In the position space representation, taking as position variable q, one has normalized eigenfunctions describing a free particle of momentum p which satisfy p = 1 e i pq P p = i d dq ( 1 e i pq 1 ) = p( e i pq ) = p p The operator Q in this representation is just the multiplication operator Qψ(q ) = q ψ(q ) that multiplies a function of the position variable q by q. The eigenvectors q of this operator will be the δ-functions δ(q q) since Q q = q δ(q q) = qδ(q q) A standard convention in physics is to think of a state written in the notation ψ as being representation independent. The wave-function in the position space representation is then taken to be the coefficient of ψ in the expansion of a state in Q eigenfunctions q, so and in particular q ψ = δ(q q )ψ(q )dq = ψ(q) q p = 1 e i pq 2

3 1.2 Momentum space representation In the momentum space description of H as functions of p, the state is the Fourier transform of the state in the position space representation, so one has 1 p = F( e i pq 1 ) = e i p q e i pq dq = 1 e i p p q dq = δ(p p ) These are eigenfunctions of the operator P, which is a multiplication operator in this representation P p = p δ(p p) = pδ(p p) The position eigenfunctions are also given by Fourier transform q = F(δ(q q )) = 1 The position operator is Q = i d dp and q is an eigenvector with eigenvalue q e i p q δ(q q )dq = 1 e i p q Q q = i d dp ( 1 e i p q 1 ) = q( e i p q ) = q q Another way to see that this is the correct operator is to use the unitary transformations F and its inverse F that relate the position and momentum space representations. Going from position space to momentum space one has Q FQ F and one can check that this transformed Q operator will act as i d dp on functions of p. Using the representation independent notation, one has p ψ = 1 ( e 1.3 Physical interpretation pq i )ψ(q )dq = ψ(p) With now both momentum and position operators on H, we have the standard set-up for describing a non-relativistic quantum particle that is discussed extensively early on in any quantum mechanics textbook, and one of these should be consulted for more details and for explanations of the physical interpretation of this quantum system. The classically observable quantity corresponding to the operator P is the momentum, and eigenvectors of P are the states that have welldefined values for this, values that will have the correct non-relativistic energy 3

4 momentum relationship. Note that for the free particle P commutes with the Hamiltonian H = P 2 2m so there is a conservation law: states with a well-defined momentum at one time always have the same momentum. This corresponds to an obvious physical symmetry, the symmetry under spatial translations. The operator Q on the other hand does not correspond to a physical symmetry, since it does not commute with the Hamiltonian. We will see that it does generate a group action, and from the momentum space picture we can see that this is a shift in the momentum, but such shifts are not symmetries of the physics. The states in which Q has a well-defined numerical value are the ones such that the position wave-function is a delta-function. If one prepares such a state at a given time, it will not remain a delta-function, but quickly evolve into a wave-function that spreads out in space. Since the eigenfunctions of P and Q are non-normalizable, one needs a slightly different formulation of the measurement theory principle used for finite dimensional H. In this case, the probability of observing a position of a particle with wave function ψ(q) in the interval [q 1, q 2 ] will be q2 q 1 ψ(q)ψ(q)dq ψ(q)ψ(q)dq This will make sense for states psi L 2 (R), which we will normalize to have norm-squared one when discussing their physical interpretation. Then the statistical expectation value for the measured position variable will be ψ Q ψ which can be computed with the same result in either the position or momentum space representation. Similarly, the probability of observing a momentum of a particle with momentumspace wave function ψ(q) in the interval [p 1, p 2 ] will be p2 p 1 ψ(p)ψ(p)dp ψ(p)ψ(p)dp and for normalized states the statistical expectation value of the measured momentum is ψ P ψ Note that states with a well-defined position (the delta-function states in the position-space representation) are equally likely to have any momentum whatsoever. Physically this is why such states quickly spread out. States with a well-defined momentum are equally likely to have any possible position. The properties of the Fourier transform imply the so-called Heisenberg uncertainty principle that gives a lower bound on the product of a measure of uncertainty in position times the same measure of uncertainty in momentum. Examples of this that take on the lower bound are the Gaussian shaped functions whose Fourier transforms were computed earlier. 4

5 For much more about these questions, again most quantum mechanics textbooks will contain an extensive discussion. 2 The Heisenberg Lie algebra In either the position or monentum space representation the operators P and Q satisfy the relation [Q, P ] = i 1 Soon after this commutation relation appeared in early work on quantum mechanics, Weyl realized that it can be interpreted as the relation between operators one would get from a representation of a three-dimensional Lie algebra, now called the Heisenberg Lie algebra. Definition. Heisenberg Lie algebra The Heisenberg Lie algebra h 3 is the vector space R 3 with the Lie bracket defined by its values on a basis (ˆp, ˆq, ĉ) by [ˆq, ˆp] = ĉ, [ˆp, ĉ] = [ˆq, ĉ] = 0 Writing a general element of h 3 in terms of this basis as pˆp + qˆq + cĉ, the Lie bracket is given by [pˆp + qˆq + cĉ, p ˆp + q ˆq + c ĉ] = (qp q p)ĉ Note that this is a non-abelian Lie algebra, but only minimally so. All Lie brackets of ĉ are zero. All Lie brackets of Lie brackets are also zero (as a result, this is an example of what is known as a nilpotent Lie algebra). The Heisenberg Lie algebra is isomorphic to the Lie algebra of 3 by 3 strictly upper triangular real matrices, with Lie bracket the matrix commutator, by the following isomorphism: pˆp + qˆq + cĉ 0 q c 0 0 p since one has 0 q c 0 q c 0 0 qp q p [ 0 0 p, 0 0 p ] = For a higher-dimensional generalization of this, one just replaces p and q by n-dimensional vectors, giving a Lie algebra h 2n+1. The physical case is n = 3, where elements of the Heisenberg Lie algebra can be written 0 q 1 q 2 q 3 c p p p

6 3 The Heisenberg group One can easily see that exponentiating matrices in h 3 gives 0 q c exp 0 0 p = 1 q c pq 0 1 p so the group with Lie algebra h 3 will be the group of upper triangular 3 by 3 real matrices with ones on the diagonal, and this group will be the Heisenberg group H 3. For our purposes though, it is better to work in exponential coordinates (i.e. identifying a group element with the Lie algebra element that exponentiates to it). Matrix exponentials in general satisfy the Baker-Campbell-Hausdorff formula, which says e X e Y = e X+Y [X,Y ]+ where the higher terms all involve commutators of commutators (one can check this term by term by expanding the exponentials, for a proof, see chapter 3 of [2]). For the case of the Heisenberg Lie algebra, since commutators of commutators vanish, we get the simpler formula for exponentials of elements of h 3 e X e Y = e X+Y [X,Y ] and we can use this to explicitly write the group law in exponential coordinates. Definition. Heisenberg group The Heisenberg group H 3 is the space R 3 with the group law (p, q, c) (p, q, c ) = (p + p, q + q, c + c (qp q p)) Note that the Lie algebra basis elements ˆp, ˆq, ĉ each generate subgroups of H 3 isomorphic to R. Elements of the first two of these subgroups generate the full group, and elements of the third subgroup are central, meaning they commute with all group elements. 4 The Schrödinger representation Since it can be defined in terms of 3 by 3 matrices, the Heisenberg group H 3 has an obvious representation on C 3, but this representation is not unitary and not of physical interest. What is of great interest is the representation (π, H = L2 (R)) of the Lie algebra h 3 on functions of q given by the Q and P operators: Definition. Schrödinger representation The Schrödinger representation of the Heisenbergy group H 3 is the representation (π, L 2 (R)) with derivative the Lie algebra representation satisfying π (ˆq)ψ(q) = iqψ(q) = iqψ(q), π (ˆp)ψ(q) = ip ψ(q) = d dq ψ(q) π (ĉ)ψ(q) = i ψ(q) 6

7 Factors of i have been chosen to make these operators skew-hermitian. They can be exponentiated, giving the unitary representation (π, L 2 (R)) of the group H 3. Elements of H 3 of the form g = e uˆq give one subgroup R H 3 and act by multiplication by a phase depending on q π (e uˆq )ψ(q) = e uiq ψ(q) = e iuq ψ(q) Those of the form g = e uˆp give another subset R H 3 and act by translation in q π (e v ˆp )ψ(q) = e vip ψ(q) = e v d dq ψ(q) = ψ(q v ) while those of the form g = e wĉ act by multiplication by a phase π (e wĉ )ψ(q) = e i w ψ(q) The group analog of the Heisenberg commutation relations (often called the Weyl form) is the relation π (e v ˆp )π (e uˆq ) = e iuv π (e uˆq )π (e v ˆp ) One can derive this by calculating (using the Baker-Campell-Hausdorff formula) π (e uˆq )π (e v ˆp ) = e iuq e ivp = e i(uq+vp )+ 1 2 [ iuq, ivp ]) uv i = e 2 e i(uq+vp ) as well as the same product in the opposite order, and then comparing the results. We have seen that the Fourier transform F and its inverse F give a unitarily equivalent representation of H 3, in terms of functions of p (the momentum space representation). The operators change as π (g) Fπ (g) F when one makes the unitary transformation to the momentum space representation. In typical physics quantum mechanics textbooks, one often sees calculations made just using the Heisenberg commutation relations, without picking a specific representation of the operators that satisfy these relations. This turns out to be justified by the remarkable fact that, for the Heisenberg group, once one picks the value of with which ĉ acts, all irreducible representations are unitarily equivalent. In a sense, the representation theory of the Heisenberg group is very simple: there s just one irreducible representation. This is very different than the theory for even the simplest compact Lie groups (U(1) and SU(2)) which have an infinity of inequivalent irreducibles labeled by weight or by spin. The unique representation of a Heisenberg group will appear in different guises (we ve seen two, will see another in the discussion of the harmonic oscillator, and there are yet others that appear in the theory of theta-functions), but they are all unitarily equivalent. This statement is known as the Stone-von Neumann theorem. 7

8 So far we ve been modestly cavalier about the rigorous analysis needed to make precise statements about the Schrödinger representation. In order to prove a theorem like the Stone-von Neumann theorem, which tries to say something about all possible representations of a group, one needs to invoke a great deal of analysis. Much of this part of analysis was developed precisely to be able to deal with general quantum mechanical systems and prove theorems about them. The Heisenberg group, Lie algebra and its representations are treated in detail in many expositions of quantum mechanics for mathematicians. Some excellent references for this material are [4], and [?]. In depth discussions devoted to the mathematics of the Heisenberg group and its representations can be found in [3] and [1]. In these references can be found a proof of (not difficult) Theorem. The Schrödinger representation described above is irreducible. and the much more difficult Theorem. Stone von-neumann Any irreducible representation of the group H 3 on a Hilbert space, satisfying π (ĉ) = i 1 is unitarily equivalent to the Schrödinger representation (π, L 2 (R)) Note that all of this can easily be generalized to the case of n spatial dimensions, with the Heisenberg group now H 2n+1 and the Stone-von Neumann theorem still true. If one tries however to take n, which is what happens in quantum field theory, the Stone-von Neumann theorem breaks down. In the case of quantum field theory one has an infinity of inequivalent irreducible representations of the commutation relations to consider, one source of the difficulties of the subject. It is also important to note that the Stone-von Neumann theorem is formulated for Heisenberg group representations, not for Heisenberg Lie algebra representations. For infinite-dimensional representations in cases like this, there are representations of the Lie algebra that are non-integrable : they aren t the derivatives of Lie group representations. For representations of the Heisenberg Lie algebra, i.e. the Heisenberg commutator relations, there are counterexamples to the Stone von-neumann theorem. It is only for integrable representations that the theorem holds and one has a unique sort of irreducible representation. 5 For further reading References [1] Folland, G., Harmonic Analysis in Phase Space, Princeton,

9 [2] Hall, B., Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer-Verlag, bibitemhall-qmbook Hall, B., An Introduction to Quantum Theory for Mathematicians, (to be published). [3] Kirillov, A., Lectures on the Orbit Method, AMS, [4] Takhtajan, L., Quantum Mechanics for Mathematicians, AMS,