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The basics of quantum mechanics and its mathematical structure. It explains the three basic axioms of quantum mechanics and how they come from a unitary representation of the Lie algebra of a Lie group. The document also defines Lie groups, Lie algebras, and unitary representations of both. It provides examples of unitary representations of Lie groups and how they relate to quantum mechanics. a lecture note and could be useful for university students preparing for an exam or studying quantum mechanics.
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Peter Woit
Columbia University
LaGuardia Community College, November 1, 2017 Queensborough Community College, November 15, 2017
Does Anyone Understand 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?
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
Three Basic Axioms of Quantum Mechanics The states of a quantum system are given by vectors ψ ∈ H where H is a complex vector space with a Hermitian inner product. In the finite-dimensional case, for vectors v, w ∈ Cn, the inner product is
v · w = v 1 w 1 + v 2 w 2 + · · · + vnwn
Observables correspond to self-adjoint linear operators on H. In the finite dimensional case these are matrices M satisfying M = M†^ = MT^. There is a distinguished observable, the Hamiltonian H, and time evolution of states ψ is given by the Schr¨odinger equation (ℏ = 1)
i
d dt ψ = Hψ
Lie groups, Lie algebras, and unitary representations
For our purposes, best to think of a Lie group G as a group of matrices, with product the matrix product. Some examples are The group SO(2) of rotations of the plane. This is isomorphic to U(1), the group of rotations of the complex plane, by ( cos θ sin θ − sin θ cos θ
∈ SO(2) ↔ eiθ^ ∈ U(1)
The group U(n) of unitary transformations of Cn. These are matrices U with U†^ = U−^1. The group GL(n, R) of all invertible linear transformations of Rn The group SO(3) of rotations of R^3. The additive group R, which can be written as matrices ( 1 a 0 1
Lie groups, Lie algebras, and unitary representations
If G is a group of matrices M, near the identity matrix we can write such group elements using the exponential as
M = eX^ = 1 + X +
The X are the elements of g. The possible non-commutativity of the group elements M is reflected in the non-zero commutator of elements of the Lie algebra. One has (Baker-Campbell-Hausdorff formula):
eX^1 eX^2 = eX^1 +X^2 +^
1 2 [X^1 ,X^2 ]+···
where [X 1 , X 2 ] = X 1 X 2 − X 2 X 1
is the commutator, and the higher order terms above can be written as iterated commutators of X 1 and X 2.
Lie groups, Lie algebras, and unitary representations
If you think of groups as “symmetries” of some object, the group is just the set of possible transformations. The object acted on together with the action of the group on it is the representation. We will be interested in “linear representations”, where the object is a vector space and the transformations are linear transformations. In particular we will take our vector spaces to be complex (Cn^ in the finite dimensional case). “Unitary” means that the transformations preserve the Hermitian inner product on Cn, so are in U(n). We have the following abstract definition: Unitary representation of a Lie group A unitary representation π on Cn^ of a Lie group G is a homomorphism π : G → U(n). This means that for every g ∈ G we have a unitary n by n matrix π(g ) ∈ U(n), and these satisfy
π(g 1 g 2 ) = π(g 1 )π(g 2 )
Lie groups, Lie algebras, and unitary representations
If G = U(n), taking π to be the identity map gives a unitary representation on Cn, the “defining representation”. There are many more possibilities. We will see that quantum mechanics produces more examples.
Lie groups, Lie algebras, and unitary representations
A unitary representation of a Lie algebra is giving us a set of linear operators π′(X ), one for each element X of the Lie algebra. These act in the finite dimensional case on a complex vector space Cn^ with Hermitian inner product. Since they are in the Lie algebra u(n) the π(X ) are skew-adjoint (π(X ))†^ = −π(X ) but if we multiply by i we get self-adjoint operators
(iπ(X ))†^ = (iπ(X ))
We see that we have self-adjoint operators acting on a complex vector space with Hermitian inner product, the same structure that appears in quantum mechanics.
Quantum Mechanics and Unitary 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 and the operators for observables should provide a unitary representation of G. This is a fertile source of interesting unitary representations of Lie groups.
Some Examples and Their Significance
Where “quantum” comes from In a very real sense, this is the origin of the name “quantum”: many physical systems have a U(1) group acting on them, so states are characterized by an integer, thus we get “quantization” of some observables. Some examples are Spatial rotations about some chosen axis, e.g. the z-axis. We find that atomic energy levels are classified by an integer: “angular momentum in z-direction.” Electromagnetic systems have a “gauge” symmetry given by a U(1) action. Such states are classified by an integer: the electric charge. The classical phase space of a harmonic oscillator has a U(1) symmetry. The quantized harmonic oscillator has states classified by an integer: the number of “quanta”.
Some Examples and Their Significance
Physical systems all come with an action of the group G = R by time translations. In quantum mechanics we expect to have a unitary representation of the Lie algebra of this group (the Lie algebra of R is also R). Such a unitary representation is given by a skew-adjoint operator which we write as −iH where H is self-adjoint. This is just the Hamiltonian operator, and the Schr¨odinger equation
d dt ψ = −iHψ
is just the statement that this operator gives the infinitesimal action of time translations. Unitary representations of G = R can, like those of U(1), be decomposed into one-dimensional representations. In this case these representations are characterized by an arbitrary real number rather than an integer. For the case of the Hamiltonian, this real number will be the energy.
Some Examples and Their Significance
The group G = SO(3) acts on physical space R^3 by rotations about the origin. Unitary representations of SO(3) break up into direct sums of irreducible components πl on C^2 l+1, where l = 0, 1 , 2 ,.. .. Physicists call these the “angular momentum l” representations. It is a major part of any course in quantum mechanics to discuss the angular momentum operators L 1 , L 2 , L 3. These are the Lie algebra representation operators coming from the fact that the quantum mechanical state space has a unitary representation of SO(3). Unlike the Pj , the operators Lj do not commute, providing a much more non-trivial example of a Lie algebra and its representations.
Some Examples and Their Significance
One can put together the two previous examples and consider the group G = E (3) of all translations and rotations of R^3. Studying the possible unitary representations of this group, one recovers essentially the usual quantum theory of a free particle moving in R^3. This generalizes to the relativistic case of four-dimensional space-time, where the symmetry group is the Poincar´e group. Its unitary representations can be decomposed into pieces which correspond to the possible quantum mechanical systems of relativistic free particles.
