Geometric formulation of quantum mechanics
Abstract
Quantum mechanics is among the most important and successful mathematical model for describing our physical reality. The traditional formulation of quantum mechanics is linear and algebraic. In contrast classical mechanics is a geometrical and nonlinear theory that is defined on a symplectic manifold. However, after invention of general relativity, we are convinced that geometry is physical and effect us in all scale. Hence the geometric formulation of quantum mechanics sought to give a unified picture of physical systems based on its underling geometrical structures, e.g., now, the states are represented by points of a symplectic manifold with a compatible Riemannian metric, the observables are realvalued functions on the manifold, and the quantum evolution is governed by a symplectic flow that is generated by a Hamiltonian function. In this work we will give a compact introduction to main ideas of geometric formulation of quantum mechanics. We will provide the reader with the details of geometrical structures of both pure and mixed quantum states. We will also discuss and review some important applications of geometric quantum mechanics.
Contents
1 Introduction
In the geometrical description of classical mechanics the states are represented by the points of a symplectic manifold which is called the phase space [Abraham_1978]. The space of observables consists of the realvalued and smooth functions on the phase space. The measurement of an observable in a state is given by . The space of observables is equipped with the structure of a commutative and associative algebra. The symplectic structure of the phase space also provides it with the Poisson bracket. An observable is associated with a vector field . Hence, flow on the phase space is generated by each observable. Moreover, the dynamics is given by a particular observer called the Hamiltonian and the flow generated by the Hamiltonian vector field describes the time evolution of the system on the phase space.
In quantum mechanics the systems correspond to rays in the Hilbert space , and the observables are represented by hermitian/selfadjoint linear operators on . Moreover, the space of observables is a real vector space equipped with two algebraic structures, namely the Jordan product and the commutator bracket. Thus the space of observables is equipped with the structure of a Lie algebra. However, the measurement theory is different compare with the classical mechanics. In the standard interpretation of quantum mechanics, the measurement of an observable in a state gives an eigenvalue of . The observable also gives rise to a flow on the state space as in the classical theory. But the flow is generated by the 1parameter group that preserves the linearity of the Hilbert space. The dynamics is governed by a specific observable, called the Hamiltonian operator .
One can directly see that these theories both have several points in common and also in difference. The classical mechanical framework is geometric and nonlinear. But the quantum mechanical framework is algebraic and linear. Moreover, the standard postulates of quantum mechanics cannot be stated without reference to this linearity. However, some researcher think that this difference seems quite surprising [Gunter_1977, Kibble_1979, Ashtekar_etal1998, Brody_etal1999] and deeper investigation shows that quantum mechanics is not a linear theory either. Since, the space of physical systems is not the Hilbert space but it is the projective Hilbert space which is a nonlinear manifold. Moreover, the Hermitian innerproduct of the Hilbert space naturally equips the projective space with the structure of a Kähler manifold which is also a symplectic manifold like the classical mechanical phase space . The projective space is usually called quantum phase space of the pure quantum states.
Let be a Hamiltonian operator on . Then we can take its expectation
value to obtain a real function on the Hilbert space which admits a
projection to . The flow is exactly the
flow defined by the Schrödinger equation on . This means that Schrödinger
evolution of quantum theory is the Hamiltonian flow on .
These similarities show us that the classical mechanics and quantum mechanics have many points in common. However, the quantum phase space has additional structures such as a Riemannian metric which are missing in the classical mechanics (actually Riemannian metric exists but it is not important in the classical mechanics). The Riemannian metric is part of underlying Kähler structure of quantum phase space. Some important features such as uncertainty relation and state vector reduction in quantum measurement processes are provided by the Riemannian metric.
In this work we will also illustrate the interplay between theory and the applications of geometric formulation of quantum mechanics. Recently, many researcher [Adler2000, Anandan1990b, Anandan1991, Gibbson1992, Marsden1999, Schilling1996, Marmo1, Marmo2, Montgomery1991, Hosh2, Levay] have contributed to development of geometric formulation of quantum mechanics and how this formulation provide us with insightful information about our quantum world with many applications in foundations of quantum mechanics and quantum information theory such as quantum probability, quantum uncertainty relation, geometric phases, and quantum speed limit.
In the early works, the most effort in geometric quantum mechanics were concentrated around understanding geometrical structures of pure quantum states and less attention were given to the mixed quantum states. Uhlmann was among the first researcher to consider a geometric formulation of mixed quantum states with the emphasizes on geometric phases [Uhlmann1986, Uhlmann(1989), Uhlmann(1991)]. Recent attempt to uncover hidden geometrical structures of mixed quantum states were achieved in the following works [GP, MB, DD, GQE, GUR, QSL].
Some researcher also argue that geometric formulation of quantum mechanics could lead to a generalization of quantum mechanics [Ashtekar_etal1998]. However, we will not discuss such a generalization in this work. Instead we concentrate our efforts to give an introduction to its basic structures with some applications.
In particular, in section 2 we review some important mathematical tools such as Hamiltonian dynamics, principal fiber bundles, and momentum map. In section LABEL:sec2 we will discuss the basic structures of geometric quantum mechanics including quantum phase space, quantum dynamics, geometric uncertainty relation,
quantum measurement, geometric postulates of quantum mechanics, and geometric phase for pure quantum states. In section LABEL:sec3 we will extend our discussion to more general quantum states, namely the mixed quantum states represented by density operators. Our review on the geometric quantum mechanics of mixed quantum states includes purification, symplectic reduction, symplectic and Riemannian structures, quantum energy dispersion, geometric uncertainty relation, geometric postulates of quantum mechanics, and geometric phase. Finally in section LABEL:sec4 we give a conclusion and an outlook. Note that we assume that reader are familiar with basic topics of differential geometry.
2 Mathematical structures
Mathematical structures are important in both classical and quantum physics. In algebraic description of quantum mechanics linear algebra and operator algebra are the most preferred structures for describing physical systems. However, in geometric quantum mechanics the most important mathematical structures are geometrical such as Hamiltonian dynamics, principal fiber bundles, and momentum maps. In this section we will give a short introduction to these topics.
2.1 Hamiltonian dynamics
In Hamiltonian mechanics the space of states or phase space is a differential manifold equipped with a symplectic form which plays an important role in describing the time evolution of the states of the system.
Here we will give a short introduction to Hamiltonian dynamics. For a detail discussion of Hamiltonian dynamics we recommend the following classical book [Abraham_1978].
Let be a smooth manifold with and be the tangent space of .
Moreover, let
(1) 
be a twoform on . Then is called symplectic if

is closed, , and

is nondegenerated, that is, for all whenever .
The pair are called a symplectic manifold. If is a smooth function on , then is a 1form on . Moreover, let be a vector field. Then we define a contraction map by . The vector field is called symplectic if is closed. Furthermore, a vector field is called a Hamiltonian vector field with a Hamiltonian function if it satisfies
(2) 
A Hamiltonian system consists of the following triple . Let be a Hamiltonian vector field. Then generates the oneparameter group of diffeomorphism
(3) 
where satisfies

with , and

for all and .
For a Hamiltonian system each point corresponds to a state of system and the symplectic manifold is called the state space or the phase space of the system. In such a classical system, the observables are realvalued functions on the phase space. Let be a function. Then is constant along the orbits of the flow of the Hamiltonian vector field if and only if the Poisson bracket defined by
(4) 
vanishes for all . Assume is a Hamiltonian vector on , and let be a point of . Moreover, let be oneparameter group generated by in a neighborhood of the point . If we assume that the initial state is , then the evolution of the state can be described by the map defined by with initial state . Under these assumptions the trajectory of is determined by the Hamilton’s equations
(5) 
Note that
(6) 
where is the Lie derivative, implies that the flow preserves the symplectic structure, that is . If the phase space is compact, then is an integral curve of the Hamiltonian vector field at the point .
Theorem 2.1.
Consider a Hamiltonian system . If is an integral curve of , then energy function is constant for all . Moreover, the flow of satisfies .
Proof.
Let be a manifold. Then an almost complex structure on is an automorphism of its tangent bundle that satisfies .
Moreover, the almost complex structure is a complex structure if it is integrable, meaning that a rank two tensor, usually called the Nijenhuis tensor vanishes [DaSilva].
Let be symplectic manifold. Then a Kähler manifold is symplectic manifold equipped with an integrable compatible complex structure. Moreover, being a Kähler manifold implies that is a complex manifold. Thus a Kähler form is a closed, realvalued, nondegenerated form compatible with the complex structure.
Example.
Let be canonical coordinate for the symplectic form, that is . Then in these coordinates we have
(8) 
where is a identity matrix.
In the following sections we will show that the quantum dynamics governed by Schrödinger and von Neumann equations can be described by Hamiltonian dynamics outlined in this section.
2.2 Principal fiber bundle
One important mathematical tool used in the geometric formulation of quantum physics is principal fiber bundles. In this section we will introduce the reader to the basic definition and properties of principal fiber bundles and in the following sections we will apply the tool to the quantum theory.
Let and be differentiable manifolds and be a Lie group. Then a differentiable principal fiber bundle
(9) 