Types of quantum field thories
While classical mechanics considers deterministic evolution of particles and fields, quantum physics follows nondeterministic evolution where the probability of various outcomes of measurement may be predicted from the state in a Hilbert space representing the possible reality: that state undergoes a unitary evolution, what means that the generator of the evolution is times a Hermitean operator called the quantum Hamiltonian or the Hamiltonian operator of the system. The theoretical framework for describing this precisely is the quantum mechanics. It involves a constant of nature, Planck constant ; some quantum systems with spatial interpretation in the limit lead to classical mechanical systems (not all: some phenomena including non-integer spin are purely quantum mechanical, but the properties depending on their existence survive in the “classical” limit); in limited generality, one can motivate and find the nonfunctorial procedure to single out a right inverse to taking this classical limit under the name quantization.
While quantum mechanics may be formulated for a wide range of physical systems, interpreted as particles, extended particles and fields, the quantum mechanics of fields is often called the quantum field theory and the quantum mechanics of systems of a fixed finite number of particles is often view as the quantum mechanics in a narrow sense.
Mathematically, despite the basic formalism of quantum mechanics which is sound and clear, there are two big areas which are yet not clear. One is to understand quantization, in all cases – of particles, fields, strings and so on. The second and possibly more central to nLab is a problem how to define rigorously a wide range of quantum field theories and some related quantum mechanical systems like the hypothetical superstring theory. Regarding that this is a central goal, we also put emphasis on the interpretation of quantum mechanics via the picture which is a special case of a FQFT, and where the time evolution functorially leads to evolution operators.
We discuss some basic notions of quantum mechanics.
It makes sense to think of this as a deformed version of a real Poisson algebra as follows:
the commutative algebra structure of the Poisson algebra coresponds to the Jordan algebra structure of the -algebra, with commutative (but non-associative!) product
With this interpretation the derivation-property of the Poisson bracket over the other product is preserved: for all we have
Given a quantum mechanical system in terms of a star algebra , we say
One can formalize the idea that a quantum mechanical system is like a deformed classical mechanical system as follows:
The -algebra canonically induces a commuative algebra ;
the (classical) states of in are in natural bijection with the quantum states externally on ;
the (classical) observables of in correspond to the external quantum observables on .
One also says that the internal classical mechanical system is the “Bohrification” of the external quantum system . See there for more details.
Given a -algebra together with a state on it, the GNS construction provides an inner product space together with an action of on and a vector – the vacuum vector? – such that for all the value of the state is obtained by applying to and then taking the inner product with :
Historically and still often in the literature, such a Hilbert space is taken as a fundamental input of the definition of quantum systems.
Traditionally, Dirac’s “bra-ket” notation is used to represent vectors in such Hilbert spaces of states, where represents a state and represents its linear adjoint. State evolutions are expressed as unitary maps. Self-adjoint operators represent physical quantities such as position and momentum and are called observables. Measurements are expressed as sets of projectors onto the eigenvectors of an observable.
In mixed state quantum mechanics, physical states are represented as density operators , state evolution as maps of the form for unitary maps , and measurements are positive operator-valued measures (POVM’s). There is a natural embedding of pure states into the space of density matrices: . So, one way to think of mixed states is a probabilistic mixture of pure states.
Composite systems are formed by taking the tensor product of Hilbert spaces. If a pure state can be written as for it is said to be separable. If no such exist, is said to be entangled. If a mixed state is separable if it is the sum of separable pure states. Otherwise, it is entangled.
As for classical mechanics, 1-parameter families of flows in a quantum mechanical system are induced from observables by
We comment on how to interpret this from the point of view of FQFT:
Quantum mechanics of point particles may be understood as a special case of the formalism of quantum field theory. It is interpreted as the quantum analog of the classical mechanics of point particles. Of course, we can take a configuration space of a system of particles looking like the configuration space of a single particle in a higher dimensional manifold.
Remark: related query on the relation between QFT and quantum mechanics (of particles and in general) can be found here.
to the cobordism of Riemannian length the operator
Such a setup describes the quantum mechanics of a particle that feels forces of backgound gravity encoded in the Riemannian metric on and forces of background gauge fields (such as the electromagnetic field) encoded in the covariant derivative .
(This is the Schrödinger picture.)
Two subsystems are called independent subsystems if the linear map
through the algebra that is generated by and inside (the smallest subalgebra containing both).
See for instance (BrunettiFredenhagen, section 5.2.2).
Given two independent subsystems , and two states and , then the corresponding product state on is defined to be
There exist states on that are not (convex combinations of) product states. This phenomenon is called entanglement.
More generally, quantum physics is all the known physics not including classical physics in wider sense; it includes relativistic and nonrelativistic phenomena. Quantum mechanics is the standard formalism with the Hilbert space, unitary evolution etc. which explains theoretically phenomena of quantum physics: in this generality of the formalism a la von Neumann, it includes the quantum field theory. Quantum statistical mechanics in fact uses some additional assumptions not in exact quantum mechanics, which are believed to be derivable eventually (like quantum ergodicity etc.). Thus quantum statistical mechanics may or may not be included within quantum mechanics.
Remark: Another way to look at quantum processes is via quantum channels which are completely positive trace-preserving maps.
For a previous query about quantum physics (includes experimental phenomena) and quantum mechanics (formalism for such, sometimes with or without statistical principles) see here.
For more on this see
The following circle of theorems
all revolve around the phenomenon that the “phase space” in quantum mechanics and hence the space of quantum states are all determined by the Jordan algebra structure on the algebra of observables, which in turn is determined by the poset of commutative subalgebras of the algebra of observables. See at order-theoretic structure in quantum mechanics for more on this.
There is also
Quantum mechanics, as opposed to classical mechanics, is necessary for an accurate description of reality whenever the characteristic scale is sufficiently small. For instance
Classical textbooks (on the Hilbert space description) include
John von Neumann, Mathematische Grundlagen der Quantenmechanik. (German) Mathematical Foundations of Quantum Mechanics. Berlin, Germany: Springer Verlag, 1932.
George Mackey, The Mathematical Foundations of Quamtum Mechanics A Lecture-note Volume, ser. The mathematical physics monograph series. Princeton university, 1963
E. Prugovecki, Quantum mechanics in Hilbert Space. Academic Press, 1971.
A list of scripts and lecture notes is at
Some standard references include
Glimm and Jaffe, Quantum physics - a functional integral point of view
P. Cartier, C. DeWitt-Morette, Functional integration: action and symmetries, Cambridge Monographs on Mathematical Physics, 2006.
Leon Takhtajan, Quantum mechanics for mathematicians , Amer. Math. Soc. 2008.
The generalization of this kind of discussion to quantum field theory is
Different incarnations of this C*-algebraic locality condition are discussed in section 3 of
relating it to the topos-theoretic formulation in
For more along these lines see the references at AQFT.