Types of quantum field thories
Dynamics affects both observables and, dually, states; this is most well known in quantum mechanics but applies equally well to classical mechanics. The different “pictures” of mechanics differ in how the dynamics is explicitly formalized:
In the Heisenberg picture, the dependence of observables on time (or more generally spacetime) is encoded, while the state is held fixed; the axiomatic formalization of this is given by the Haag–Kastler axioms of AQFT.
The Dirac (interaction) picture is a mixture of these two approaches: dynamics is split into a free? (or otherwise solvable?) part and an interaction or perturbation; one of these is take to affect the states, the other the observables.
Let us assume a global notion of time, say a fixed background? spacetime which is globally hyperbolic, so that it admits a foliation into Cauchy surfaces, and choose a time coordinate? for this foliation. The upshot of this is that each event occurs at a time , and conversely we can speak of space at any time (at least within certain bounds). Thus we may speak sensibly of either the state of the world at time or the value of some observable quantity at time .
Because this is a picture of dynamics, states or observables (as appropriate to the picture) will vary through time. We therefore have a time evolution? operator between any two times ; actually, we need consider only , since .
In the Heisenberg picture, for each observable , we speak of only at some time , so our actual observables are of the form . We write abstractly
to show the evolution of the observable through time. However, when it comes to the state of the world, we speak of a single state that describes the world at all times.
In the Schrödinger picture, we instead speak of , the state of the world at time . We write abstractly
(the Schrödinger equation) to show the evolution of the state through time. However, when it comes to observables, we use only the observable across all times.
To see the connection between the two pictures, recall that an observable and a state together produce a probability distribution giving the probability that any given value of will be observed, given that the world is in state . (This is true throughout mechanics, although it is obscured in non-statistical classical mechanics, since the probability distributions produced by classical pure states are all delta measure?s.) Assuming that belongs to an appropriate algebra of observables and the probability measures are sufficiently nice, we may restrict attention to the expectation values of these distributions, since the entire distribution can be recovered from as varies over natural numbers.
The connection between the two pictures is then given by
It remains to say exactly what is and what the operations and are. Let us use the density matrix formulation of quantum statistical mechanics?, since classical and non-statistical mechanics may be recovered as special cases, by restricting (respectively) the allowed observables or states. In this case, both states and observables are given by linear operators on a Hilbert space , and we have
(a right action?) and
(a left action?). We then have
as desired, using the cyclic property of the trace?.
The time evolution operator is often derived from a Hamiltonian and the formula for or is further derived from a differential equation involving this Hamiltonian. However, this is unnecessary for the connection between the two pictures.
If spacetime is not globally hyperbolic, then there is no time coordinate , and none of the discussion above makes sense; or if we choose a coordinate and call it time regardless, then time evolution is not a symmetry and we do not have the operators .
In this case, the Heisenberg picture still makes sense, even though we cannot expect to calculate from (if it even makes sense to discuss such things). This is easily seen in field theory, where the operators called above are really of the form . Then the Heisenberg picture's is really , or simply where indicates an event (a point in spacetime). So even if the coordinates do not make sense, still does; and even if the equations of physics cannot be thought of as describing evolution through time, still they can be thought of as describing the relationships between observables at different places in spacetime.
In contrast, the Schrödinger picture cannot be so treated. One may be led to the contrary impression by the quantum mechanics of a single particle without any internal structure (not even spin), in which case the Hilbert space of (pure quantum-mechanical) states is naturally identified with and the state is really . In this case, the Schrödinger picture's is really , that is . However, this fails in classical or statistical mechanics; and even in non-statistical quantum mechanics, it breaks down if the particle has internal structure or there is more than one particle in the world. Then we see that the spacial coordinates generalise to the arbitrary coordinates of configuration space, while remains only , and there is no way to subsume it into a spacetime coordinate.
The Heisenberg picture is the only one where relativistic quantum field theory can be rigorously defined, e.g., through the Wightman axioms. The latter imply the existence of a 4-dimensional group of translations generated by the 4-momentum vector . Once one select a future-like direction to determine time, one gets from the Hamiltonian , and can use it to define time-dependent states in the usual way. Together with the spatial part of the momentum, this gives a conventional, frame-dependent Schroedinger representation of the states.
Thus the Schroedinger picture exists but is frame dependent. You can regard any operator on the Hilbert space as a Schroedinger observable and find its expectation as time changes in the usual Schroedinger way, and translate it to the Heisenberg picture in the usual way such that becomes explicitly time-dependent and the state is fixed.
However, in 4-dimensional relativistic interacting quantum field theories, fields must be smeared in space and in time in order to produce densely defined operators (rather than distributions). Thus, unlike in the nonrelativistic case, the fixed-time spatial fields , where is 3-dimensional, are not well-defined objects.
Thus although the Schroedinger picture exists it does not represent local fields at a fixed time. The attempt to pretend it did leads to the problems mentioned in Dirac’s paper. In this sense, the relativistic field theory cannot be made well-defined without transcending the Schroedinger picture.
The above is completely independent of scattering theory. In scattering theory one has to construct the asymptotic Hilbert space. Unlike the interacting Hilbert space, the asymptotic Hilbert space is a Fock space and must therefore be defined by a limiting procedure. This is what is done in the Haag-Ruelle theory. Note that because of the Lorentz invariance of the future cone, the resulting asymptotic space does not depend on the choice of the time direction, as long as it points into the future cone.
There are presentations of the Haag-Ruelle theory that are a little more in the spirit of the Heisenberg picture; see the comments at the end of p.379 of Volume 3 of Reed and Simon. One can probably work almost completely in the Heisenberg picture if one works out the algebra of asymptotic constants (in analogy with the nonrelativistic case treated in Section 3.4 of Volume 3 of Thirring), but to show that the corresponding asymptotic operators have a Fock representation one apparently needs to go through some Schroedinger-like computations.
Historically, the terms ‘Schrödinger picture’ and ‘Heisenberg picture’ (at least) referred to more than what we discuss above; they referred to the entirety of the differences between Schrödinger's and Heisenberg's approaches to quantum mechanics.
For example, these terms included also Schrödinger's use of typically wave-like functions as pure states (and correspondingly operators in the higher-type-theoretic sense as observables) vs Heisenberg's use of infinite-dimensional matrices as observables (and correspondingly infinite sequences as pure states). This difference was rectified by von Neumann's application of Hilbert space to the problem, showing that (if one suitably restricts the allowed functions and sequences and also identifies equivalent functions a bit) both approaches used Hilbert space (what we would now call the infinite-dimensional separable Hilbert space) as the space of pure states.
This is entirely separate from the question of whether states or observables are taken to evolve with time. Still, there is this connection: Schrödinger evolved states, and his approach was called ‘wave mechanics’ after his representation for states, while Heisenberg evolved observables, and his approach was called ‘matrix mechanics’ after his representation for observables.
|Poisson algebra||Poisson manifold|
|deformation quantization||geometric quantization|
|algebra of observables||space of states|
|Heisenberg picture||Schrödinger picture|
|higher algebra||higher geometry|
|Poisson n-algebra||n-plectic manifold|
|En-algebras||higher symplectic geometry|
|BD-BV quantization||higher geometric quantization|
|factorization algebra of observables||extended quantum field theory|
|factorization homology||cobordism representation|
See for instance sections 7.19.1–3 in
To check conventions at least, see Wikipedia: