In this post, I want to explain briefly the idea behind AQFT in the Haag-Kastler style. To motivate this, let me first sketch what QFT (= quantum field theory) is about, at least in my mathematically distorted perception.

Classical quantum theory is about modelling purely quantum effects, i.e. without considering gravity, or at least without considering relativistic effects. There, a separable Hilbert space as a state space is appropriate. The bounded linear operators on the state space form a certain kind of normed algebra with compatible involution (taking the adjoint) called C*-algebra and measurements correspond to self-adjoint operators.

Quantum field theory tries to incorporate quantum mechanics into electromagnetic field theory (or vice versa) and gravitational field theory. So far, no such theory-of-everything has been developed with falsifiable predictions, although there are some promising candidates.

Continue reading «Some thoughts on AQFT (algebraic or axiomatic quantum field theory)»

See also: Questions part I - Information and Entropy
Questions part II - Complexity and Algorithmic Complexity

Questions part III - Statistical Physics, Quantum Physics and Thermodynamics [GM94] [Pen05]

1. Can there be some kind of Maxwell’s daemon, and why (not)?
2. Can all analog information be transformed to digital information losslessly?
   Is physical information always digital (digital philosophy)?
3. Does the second law of thermodynamics depend on a human observer?
4. Can quantum mechanical entanglement be reduced to mutual information? [NC00] [DH00] [PV06] [BP07]
5. What is the mathematical, what is the physical content of Heisenberg’s uncertainty relation?
   

   “I think I can safely say that nobody understands quantum mechanics.” – Richard Phillips Feynman, 1965

6. Is there a method to denote and calculate, for an open physical system, how the information content changes in time when the system’s dynamics are known?
7. Is there an important difference between information loss into small scales and information loss into microscopic freedoms?
8. Where do (thermal, quantum, vacuum, ?) fluctuations come from? [Lam97] [Lam98]
   What do they change on the macroscopic scale?
   Are gravitational fluctuations possible?
9. According to Einstein’s fluctuation-dissipation-theorem, do thermal fluctuations compensate exactly for information loss through dissipation? [CW51]
10. Is probability theory powerful enough to capture any micro-physical phenomena?
    Is mathematical probability theory the correct language for modern physics? [Mil04]

    “In fact the smallest units of matter are not physical objects in the ordinary sense; they are forms, ideas which can be expressed unambiguously only in mathematical language.” – Werner Heisenberg, 1992

11. How is Zeilinger’s concept of elementary systems generalizable to general state spaces?
12. Given the fact that every system is continually measured, is the concept of a closed quantum system (with unitary time evolution) relevant for real physics? [And97]
    Does decoherence by interference with the background universe render the concept of closed quantum systems obsolete? [BTV09]
13. How does randomness in quantum measurement emerge from unitary evolution?
    Is quantum physics truly information-preserving?
14. How relevant is the classical concept of degree of freedom for quantum mechanics?

References

• [And97] Philip W. Anderson, Is measurement itself an emergent property?, Complex. 3 (1997), no. 1, 14–16.
• [BP07] S.L. Braunstein and A.K. Pati, Quantum information cannot be completely hidden in correlations: Implications for the black-hole information paradox, Physical Review Letters 98 (2007).
• [BTV09] Buchleitner, Tiersch, and Viviescas, Entanglement and decoherence, Lecture Notes in Physics, Springer, 2009.
• [CW51] Herbert B. Callen and Theodore A. Welton, Irreversibility and generalized noise, Phys. Rev. 83 (1951), no. 1, 34–40.
• [DH00] D. Deutsch and P. Hayden, Information flow in entangled quantum systems, Proceedings: Mathematics, Physical and Engineering Sciences 456 (2000), no. 1999, 1759–1774.
• [GM94] M. Gell-Mann, The quark and the jaguar, Freeman New York, 1994.
• [Lam97] SK Lamoreaux, Demonstration of the casimir force in the 0.6 to 6 µm range, Physical Review Letters 78 (1997), no. 1, 5–8.
• [Lam98] S. K. Lamoreaux, Erratum: Demonstration of the casimir force in the 0.6 to 6 µm range [phys. rev. lett. 78, 5 (1997)], Phys. Rev. Lett. 81 (1998), no. 24, 5475–5476.
• [Mil04] David Miller, Probability generalizes logic, 2004.
• [NC00] Nielsen and Chuang, Quantum computation and quantum information, 2000.
• [Pen05] Roger Penrose, The road to reality: A complete guide to the laws of the universe, 3 ed., Knopf, 2005.
• [PV06] Martin B. Plenio and S. Virmani, An introduction to entanglement measures, arXiv, Jun 2006.