Schrodinger, Heisenberg, Feynman...there is more than one formal development of quantum physics. The problem of this day is integrating quantum physics with gravitational physics. The Algebraic construction described by Robert Wald in a recent paper could be a fourth such formalism. The well constructed theories of quantum gravity conform to what he wrote in his paper (or chronologically his paper inadvertently describes the general formalism that has gone into LQG, String/M-theory, and even my little theory of quantum gravity).
There are a number of axiomatizeations of quantum theory. Schrodingers wave mechanics, Heisenberg's matrix mechanics, and Feynman's path integral formulation (which while used mostly for interacting field theory can be used for quantum theory as well) and some physicist such as the man I learned quantum theory from like to mix their own set of axioms. The following construction could be used for all of quantum mechanics but I will focus on quantum gravity.
Wald's algebraic construction:
I have worked on quantum gravity independently for quite some time. I have also dabbled in traditional quantum field theory in curved space time. In loop quantum gravity, M(atrix) theory, and my theory, each proposes what seem to be very different things. Then I read a paper by Robert Wald of the university of Chicago "The Formulation of Quantum Field Theory in Curved Spacetime" arXiv:0907.0416v1. In which he proposes the following.
The algebraic approach: In the algebraic approach, instead of starting with a Hilbert space of states and then defining the field observables as operators on this Hilbert space, one starts with a *-algebra, A, of field observables. A state, ω, is simply a linear map ω : A → C that satisfies the positivity condition ω(A∗A) ≥ 0 for all A ∈ A. The quantity ω(A) is interpreted as the expectation value of the observable A in the state ω. If H is a Hilbert space which carries a representation, π, of A, and if then the map ω : A → C given by
defines a state on A in the above sense.Dr. Wald then explains how a Hilbert space representation of A can be found by way of the Gelfand–Naimark–Segal construction. He states that
The key difference is that, by adopting the algebraic approach, one may simultaneously consider all states arising in all Hilbert space constructions of the theory without having to make a particular choice of representation at the outset. It is particularly important to proceed in this manner in, e.g., studies of phenomena in the early universe, so as not to prejudice in advance which states might be present.This statement makes me think of the various theories of quantum gravity.
In his paper he goes on to take on the remaining assumptions that go into quantum field theory in flat space-time and postulate replacements for each of them.
- Spectrum condition --->micro local spectrum condition
- Global poincare invariance ---> local poincare invariance*
- "...the condition that at spacelike separations quantum fields either commute or anticommute generalizes straightforwardly to curved spacetime."
I am going to say that this construction of quantum field theory in curved space time is more than just a QFT for a fixed curved space-time. Wald has set out a recipe for the quantization of space-time itself, which is what is done in two seemingly incompatible ways in Loop Quantum Gravity, and in my theory Quantum Space-time dynamics. Even string theory could likely be formulated in a way which would conform to Walds recipe! All any fundamental theory of physics is just a algebra of field observables. Once one has an acceptable algebra of field observables the rest follows naturally if one knows of this.
Examples LQG and Quantum Space-Time Dynamics:
According to Carlo Roveli Loop Quantum Gravity is a algebra of fields defined on a manifold one of which is space-time.
One can still describe spacetime as a (differentiable) manifold (a space without metric structure), over which quantum fields live. A classical metric structure will then be defined only by expectation values of the gravitational field operator.With the algebra of the field being a loop algebra.
I wrote in my book on Quantum Space-Time Dynamics about an operator S the eigenvalues of which are equivalent to possible metrics, s, in general relativity. The metric does not make an appearance in the theory until that point. It is derived from the algebra of the operators which I discovered after I had worked out the physical consequences of the theory as being the lie group and algebra F(4). It could be more orthodoxly formulated in terms of Hurwitz quanternions. However the representation seen in my book, I feel, has the advantage of seeming familiar and being more physically intuitive.
I am certain that string/M theory can also be shown to conform to Dr. Walds general prescription.
Wald has done something that I suspected was possible, that LQG theorist have thought was possible, he has provided the framework in which non string quantum gravity and string theory can look the same. Perhaps LQG and string theory could be unified using this construction. LQG/or my theory describing gravity and space time, String theory describing every thing else. My own theory already has room for this because in this theory "everything else" already has an operator T to describe it.
A theory of quantum gravity and possibly everything else that may be in the universe is close at hand.