# Noncommutative Spacetime; Rindler; Liouville; Strings (15)

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

**Page 1, dated 6/16/08: notations here are a continuation from the previous notebook ending 6/15/08 (still to be uploaded).

(p. 43) There is no local conservation, only global? Let’s turn this problem into an advantage… So, we can choose the Moyal Brackets to preserve local conformal invariance! (see note on p.72 below).

(p. 45) The complexities encountered concerning the correct interpretation of the conservation equation div t=0 seems strongly depend on the definition given to conserved currents It suggests 2 strategies: 1) Mod out by total commutation. I.e. equivalence relation f ~ g if f=g+[ , ] M . 2) Keep everything under integral signs. →→ no local conservation, only global. This may not be consistent with Poincaré invariance.

(p. 47) Returning to the question of why SUR’s require extended dynamical objects with internal DOF’s.

(p. 65) The Moyal Bracket term has the property that its integral vanishes Greenberg has demonstrated violation of microcausality (this is just nonlocality). Commutators of fee fields do not vanish at spacelike separations The laws of physics should be consistent They should produce only finite results for observables. One can see why Q.M. is necessary in statistical mechanics, because otherwise the entropy of a finite amount of gas is infinite. With Q.M., the entropy is proportional to ( v/h3 ), where v is a phase space volume. What determines the size of h?

(p. 66-67) Some issues should be clarified concerning Noether’s Theorem in the non-commutative case. 1) do conserved charges generate symmetries? In particular, the question arises in theories with finite nonlocality over a distance ξ. 2) Does time play a role different from space in these considerations? Do theories noncommutative in space behave differently from those non-commutative in time on the classical level of Hamiltonian/Lagrarian dynamics? Examine higher derivatives and locality along the lines of the two questions on this page. Work out the Hamiltonian structure of the nonlocal interacting theory in Fujikawa.

(p. 72-73) We cannot “Mod out by total Mayol Brackets” because this just yields the commutation algebra!…And the theory becomes the commutative one with a fancy multiplication sign! Quantum Theory regularizes the space of states, by using noncommutative geometry. What is the classical space of states of the gravitational field?

(p. 92) A Circle of Ideas.

A-circle-of-ideas

• (p. 101-107) Topics covered:
• General Relativity is a theory of spacetime structure and gravitation. The theory has many solutions and well defined initial value formulation. –Some solutions: 1) Black hole with a metric that becomes asymptotically symmetries (Flat, dS, AdS) 2) FRW cosmology 3) Binary star system emitting energy in the form of gravitational radiation. N.B. That asymptotic structures directly affects local spacetime physics, e.g. spectrum of bh depends on asymptotic conditions. The solutions must be considered as a whole in order for exact considerations to be carried out. Cosmology must be considered as directly affecting local particle physics. This is almost action at a distance–what happens if we adjust this asymptotes? How does a local observer receive this signal?
• Holographic Spacetime
• Covariant Entropy (bound)
• What is the Hilbert Space of Dpa? This is where new holographic physics comes in.
• Proposal: Discretize the holographic screen without breaking its symmetries→→Noncommutative Geometry
• Open-Closed duality–Proposition that open and closed string theories are dual.
• (p. 108-110) Noncommutativity and Conformal Symmetry
• Spacetime noncommutativity in the tight context of SDYM. What do the conserved charges and currents mean? The theory is nonlocal. Continuity equation seen to be the epitome of local physical construction. How are these ideas made compatible? What does it mean, in essence, for a theory to be nonlocal? Of course, from the physical point of view it suggests that only when entities are in ‘proximity’ can they interact. The traditional notion of proximity is: interaction at a point.
• Non locality may be physically equivalent to the dynamics of extended objects. THIS may be the reason why String Theory contains gravity.
• Something absolutely magical seems to happen when quantum dynamics is applied to extended objects.

(p. 119) S-Matrix = Indirect knowledge (perhaps the only kind of knowledge?) We humans observe correlations between the 1 in > and 1 out > states and imagine a mechanism { = ( ?? ) } that correctly reproduces the correlations. The mechanism is then assumed to govern physical processes in the interaction regions = ( ). How to extract spacetime uncertainties from scattering amplitudes? Does the S-Matrix approach to spacetime structures really make sense?

(p. 125) In quantum theory one always detects, in a measured process, a quantum as existing at a point. Well, sort of…Geiger counters respond with ‘clicks‘, as photographic plates respond with darkened ‘spots‘. The mental image suggested is that of a point particle with a probability of existing in places where the wave function has support. This picture must be fundamentally inadequate.