Quantum Mechanics and Representation Theory, Slides of Quantum Mechanics

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Quantum Mechanics and Representation Theory

Peter Woit

Columbia University

Texas Tech, November 21 2013

Does Anyone Understand Quantum Mechanics?

“No One Understands Quantum Mechanics”

”I think it is safe to say that no one understands quantum mechanics” Richard Feynman The Character of Physical Law, 1967

Does Anyone Understand Quantum Mechanics?

Outline

Today would like to Explain how Lie groups and unitary representations are related to quantum mechanics, providing some sort of “understanding” of the structure of the subject. Advertise a book project in progress, based on a course taught last year. Hope to finish draft early next year, about 80 percent done, see my web-site at Columbia. Explain some unanswered questions about representation theory raised by physics. Quickly indicate some work in progress: possible relevance of some new ideas in representation theory (“Dirac cohomology”) to physics.

Does Anyone Understand Quantum Mechanics?

What We Really Don’t Understand About Quantum

Mechanics

While representation theory gives insight into the basic structure of the quantum mechanics formalism, a mystery remains

The mystery of classical mechanics We don’t understand well at all how “classical” behavior emerges when one considers macroscopic quantum systems.

This is the problem of “measurement theory” or “interpretation” of quantum mechanics. Does understanding this require some addition to the fundamental formalism? Nothing to say today about this.

Quantum Mechanics

Where do these axioms come from?

A unitary representation of a Lie group G gives exactly these mathematical structures: A complex vector space V = H, the representation space. Differentiating the representation homomorphism π at the identity, one gets a “Lie algebra homomorphism”

π′^ : X ∈ Lie(G ) = Te G → π′(X )

The condition that the π(g ) be unitary implies that the π′(X ) are skew-Hermitian. Multiplying by i, the iπ′(X ) are Hermitian linear operators on V.

Quantum Mechanics

Two-dimensional quantum systems, I

For the simplest non-trivial example of this, take a quantum system with H = C^2 (called the “qubit”). The Lie group U(2) of two-by-two unitary (U−^1 = (U)T^ ) matrices acts on H by the defining representation (π the identity map, π′^ also the identity). Such matrices are of the form

U = etX

where X is skew-Hermitian (X

T = −X ). The Lie algebra of U(2) is a four-dimensional real vector space, with basis {i 1 , iσ 1 , iσ 2 , iσ 3 }, where the σj are Hermitian matrices, the physicist’s “Pauli matrices”:

σ 1 =

, σ 2 =

0 −i i 0

, σ 3 =

Quantum Mechanics and Unitary Representations

Relating Quantum Mechanics and Representations

Basic Principle Quantum mechanical systems carry unitary representations π of various Lie groups G on their state spaces H. The corresponding Lie algebra representations π′^ give the operators for observables of the system.

Significance for physicists Identifying observables of one’s quantum system as coming from a unitary representation of a Lie group allows one to use representation theory to say many non-trivial things about the quantum system.

Significance for mathematicians Whenever physicists have a physical system with a Lie group G acting on its description, the state space H should provide a unitary representation of G. This is a fertile source of interesting unitary representations of Lie groups.

Quantum Mechanics and Unitary Representations

Example: translations in space, G = R^3

Physics takes place in a space R^3. One can consider the Lie group G = R^3 (the “space translation group”). The quantum state space H will provide a unitary representation of this group. The Lie algebra representation operators are called the “momentum operators” Pj , j = 1, 2 , 3

These commute, so can be simultaneously diagonalized, in a basis of H of states called the “momentum basis.” Basis elements have well-defined values for the components pj of the momentum vector (the eigenvalues of the operators Pj ). By Heisenberg uncertainty, these states are not “localized”, carry no information about position.

Quantum Mechanics and Unitary Representations

Example: Quanta, G = U(1)

Many physical systems have a group G = U(1) (circle group of phase rotations) acting on the system. Any unitary representation of U(1) is a direct sum of one-dimensional representations πn on C where

πn(eiθ) = einθ

for n ∈ Z. The Lie algebra representation is given by an anti-Hermitian operator with eigenvalues in, n ∈ Z. The physicist’s Hermitian observable operator is called N, and has eigenvalues n.

Where “quantum” comes from In a very real sense, this is the origin of the name “quantum”: many systems have a U(1) group acting, so states are characterized by an integer, which counts “quanta”. Sometimes this has an interpretation as “charge”.

Quantum Mechanics and Unitary Representations

Example: Rotations, G = SO(3) or G = SU(2)

The group G = SO(3) acts on R^3 by rotations about the origin, and it has a double cover G = SU(2). Unitary representations of SU(2) break up into direct sums of irreducible components πn on Cn+1, where n = 0, 1 , 2 ,... (for n even these are also representations of SO(3)). Physicists call these the “spin n 2 ” representations. Recall that a basis of the Lie algebra of SU(2) is given by {iσ 1 , iσ 2 , iσ 3 }. The corresponding observables are the operators

Jj = −iπ′ n(iσj )

These are called the “angular momentum operators” for spin n 2. They do not commute and cannot be simultaneously diagonalized.

Symmetries

Lie Group Representations and Symmetries

When the action of a Lie group G on a quantum system commutes with the Hamiltonian operator, G is said to be a “symmetry group” of the system, acting as “symmetries” of the quantum system. Then one has

Conservation Laws Since the observable operators O corresponding to Lie algebra elements of G commute with H, which gives infinitesimal time translations, if a state is an eigenstate of O with a given eigenvalue at a given time, it will have the same property at all times. The eigenvalue will be “conserved.”

Degeneracy of Energy Eigenstates Eigenspaces of H will break up into irreducible representations of G. One will see multiple states with the same energy eigenvalue, with dimension given by the dimension of an irreducible representation of G.

Symmetries

Lie Group Representations Are Not Always Symmetries

The state space of a quantum system will be a unitary representation of Lie groups G , even when this action of G on the state space is not a symmetry, i.e. doe not commute with the Hamiltonian. The basic structure of quantum mechanics involves a unitary group representation in a much more fundamental way than the special case where there are symmetries. This has to do with a group that already is visible in classical mechanics. This group does not commute with any non-trivial Hamiltonian, but it plays a fundamental role in the theory.

Quantization

The Group of Canonical Transformations

Sophus Lie first discovered Lie groups in the context of Hamiltonian mechanics. It turns out that there is an infinite-dimensional group acting on a phase space M, known to physicists as the group of “canonical transformations”. For mathematicians, this is the group G = Symp(M) of symplectomorphisms, the group of diffeomorphisms of M preserving the symplectic form, a two-form ω on M given by

ω =

∑^ n

j=

dqj ∧ dpj

It turns out that the Lie algebra of this group is given by the functions on M, with the Poisson bracket giving the Lie bracket, the structure which reflects the infinitesimal group law near the identity of the group. The Poisson bracket has the right properties to be the Lie bracket of a Lie algebra: it is anti-symmetric, and satisfies the Jacobi identity.

Quantization

First-year Physics Example

A classical particle of mass m moving in a potential V (q 1 , q 2 , q 3 ) in R^3 is described by the Hamilton function h on phase space R^6

h =

2 m

(p 12 + p 22 + p 32 ) + V (q 1 , q 2 , q 3 )

where the first term is the kinetic energy, the second the potential energy. Calculating Poisson brackets, one finds

dqj dt

= {qj , h} =

pj m

=⇒ pj = m

dqj dt and dpj dt

= {pj , h} = −

∂V

∂qj where the first equation says momentum is mass times velocity, and the next is Newton’s second law (F = −∇V = ma).