There are various more or less formal ways of expressing the foundational principles of nonrelativistic quantum mechanics, including both nonmathmatical statements and more rigorous axiomatizations. Recently there have been various people who have treated this from a quantum-computing point of view. Others have explored whether QM can be bent without breaking it. I've given some references at the bottom of this question to some of this kind of work.
But more concretely, I think most physicists would consider the following to be some sort of consensus on an informal list of principles. (Actually, I'd be happy to hear criticisms of this list as well.)
- Wavefunction fundamentalism. All knowable information about a system is encoded in its wavefunction (ignoring phase and normalization).
- Unitary evolution of the wavefunction. The wavefunction evolves over time in a deterministic and unitary manner.
- Observables. Any observable is represented by a Hermitian operator.
- Inner product. There is a bilinear, positive-definite inner product on wavefunctions.
- Completeness. For any system of interest, there exists a set of compatible observables such that any state of the system can be expressed as a sum of eigenstates.
Question: Does this summary of principles have to be modified for QFT? If so, how? If not, then what is the core difference between these two theories?
Mackey, The Mathematical Foundations of Quantum Mechanics, 1963, p. 56ff
Aaronson, "Is Quantum Mechanics An Island In Theoryspace?," http://arxiv.org/abs/quant-ph/0401062
Masanes and Mueller, "A derivation of quantum theory from physical requirements," https://arxiv.org/abs/1004.1483
Hardy, "Quantum Theory From Five Reasonable Axioms," https://arxiv.org/abs/quant-ph/0101012
Dakic and Brukner, "Quantum Theory and Beyond: Is Entanglement Special?," https://arxiv.org/abs/0911.0695
Banks, Susskind, and Peskin, "Difficulties for the evolution of pure states into mixed states," Nuclear Physics B, Volume 244, Issue 1, 24 September 1984, Pages 125-134
Nikolic, "Violation of unitarity by Hawking radiation does not violate energy-momentum conservation," https://arxiv.org/abs/1502.04324
Unruh and Wald, https://arxiv.org/abs/hep-th/9503024
Ellis et al., "Search for violation of quantum mechanics," Nucl Phys B241(1984)381
Gisin, "Weinberg's non-linear quantum mechanics and supraluminal communications," http://dx.doi.org/10.1016/0375-9601(90)90786-N , Physics Letters A 143(1-2):1-2
Sebens and Carroll, "Self-Locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics," https://arxiv.org/abs/1405.7577