quantum mechanics - Differences between principles of QM and QFT

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.)

1. Wavefunction fundamentalism. All knowable information about a system is encoded in its wavefunction (ignoring phase and normalization).


2. Unitary evolution of the wavefunction. The wavefunction evolves over time in a deterministic and unitary manner.


3. Observables. Any observable is represented by a Hermitian operator.


4. Inner product. There is a bilinear, positive-definite inner product on wavefunctions.


5. 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?

References

Kapustin, https://arxiv.org/abs/1303.6917

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

Answer

1. 
   

   All knowable information about a system is encoded in a ray in a Hilbert space. In QFT, and unlike non-relativistic QM, there is no $|x\rangle$ basis, so you cannot construct a wave-function $\varphi(t,x)=\langle x|\varphi(t)\rangle$ to encode this information. What you can do is encode this information in the so-called correlation functions (cf. Wightman Reconstruction Theorem). You need an infinite number of functions to encode all the information of the system. Equivalently, one may encode this same information in a single functional, either through a functional integral or as a wave functional (cf. 214552).

   
   


2. 
   

   This is unchanged, except perhaps for the fact that it is usually much more convenient to evolve operators instead of states, because covariance becomes manifest. The abstract Schrödinger equation, $\frac{\mathrm d}{\mathrm dt}|\psi\rangle=-iH|\psi\rangle$ is as valid in non-relativistic QM as it is in QFT (and so is the Heisenberg equation, $\dot A=i[H,A]$). In this sense, the evolution is still unitary, but it is expressed in terms of operators instead of states.

   
   



3. 
   

   This is unchanged.

   
   


4. 
   

   This is unchanged, except perhaps for the fact that it is sometimes convenient to artificially enlarge the Hilbert space so as to include "negative norm states", that is, the inner product is relaxed into a sesquilinear form (which agrees with the positive-definite "true" inner product in the "true", physical Hilbert space).

   
   


5. 
   

   This is unchanged.