# Noncommutative Spacetime & Rindler Space (18)

Below: A few notations to indicate topics covered. Author’s page numbering is ignored, page numbers referenced below are from the scanned PDF documents.

Notebook: August 2005 – January 2006 https://gaugegravity.com/wp-content/uploads/2023/06/Vol.-8-8-2005-1-2006-NC-Spacetime.pdf

(p. 47-48) The “Passive Observer,” exemplified most clearly by “coordinate transformations” must be banished from physics! Every observer corresponds to a non zero energy density at least. The observer’s existence changes the problem. In the spirit of incorporating the observer into the system we can, we must phrase question in terms of “probe observers.”

• (p. 74 refer to pdf for ellipses) The Rindler Space Project
• What I have done so far:
• A. Derived the Algebra of Rindler Space
• B. Written an action that yields sensible classical equations of motion upon variation.
• What I still have to do:
• A. Calculate the path integral based on the action above…
• What is giving me fits is the appearance of star products in the Kinetic terms! I have to find the inverse of…. To do this I should go back to the operator POV, which can be found in Lizzi.
• B. Analyze unitarity – Cutting rules, Gomis, Mehen
• Is there any way to recover unitarity? What about the modified T-ordering?
• C. Hamiltonian Analysis
• This could be a check on the Path Integral, but might not be necessary.
• D. Unruh Effect – Is there a path integral derivation of the Unruh Effect? (If not, part C is essential)

(p. 111-114) Let’s work on the path integral. So, the upshot seems to be that only the first order contributes. Higher orders can be made to vanish via integration by parts.

(p. 117) Boundary Terms: Different forms of the action, differing by a boundary term yield different boundary conditions. From page 30 (pdf p.60).

• (p. 120-124) Let’s put the path integral aside for the moment and think about the general validity of space-time noncommutative Field Theory.
• The Theory is nonunitary and, therefore, can at best describe part of an unstable system. If we follow the string theory argument, oscillator DOF’s of the strings will be excited, even if they are initially unexcited. (embed below)
Strings

(p. 129) The fact that spacetime noncommutative field theory does not exist may be a major clue into the nature of string theory. The connection between unitarity/causality and analyticity/locality may be illuminated.

• (p. 135) Path integral quantization can be preformed – See Fugikawa’s paper but, while the usual cutting rules can be restored, it is only at the expense of tachyonic states with a sick dispersion relation.
• Time – NCFT is describing part of an unstable physical system. The system is unstable because of tachyonic modes in the asymptotic Hilbert space. These modes are remnants of closed string states that did not decouple when the “low energy limit” was taken. Closed strings are associated with gravitation and spacetime, so instability of closed string modes implies an instability associated with gravitation or an instability of the underlying spacetime itself. Could the rigged Hilbert space formalism be applied?

(p. 139 – 169) Some thoughts on Casimir Dark energy for noncommutative internal spaces.

(p. 169 – ) Path Integrals in the presence of boundaries and Horizons.

Included below are two sets of notes which were inserted at the conclusion of this notebook. The first set was dated 09-10 2006, titled Non Comm. Field SP and the second was dated 01-02 2007.