About the origin of Born rule
Lubos has been again aggressive. At this time Sean Carroll became the victim of Lubos's verbal attacks. The reason why Lubos got angry was the articles of Carroll and his student to derive Born rule from something deeper: this deeper was proposed to be many worlds fairy tale as Lubos expresses it. I agree with Lubos about the impossibility to derive Born rule in the context of wave mechanics - here emphasis is on "wave mechanics". I also share the view about many worlds interpretation - at least I have not been able to make any sense of it mathematically.
Lubos does not miss the opportunity to personally insult people who tell about their scientific work on blogs. Lubos does not realize that this is really the only communication channel for many scientists. For the out-laws of the academic world blogs, home pages, some archives, and some journals (of course not read the "real" researchers enjoying monthly salary) provide the only manner to communicate their work. Super string hegemony did good job in eliminating people who did not play the only game in the town: also I had the opportunity to learn this.
Ironically, also Lubos is out-of-law, probably due to his overly aggressive blog behaviors in past. Perhaps Lubos does not see this as a personal problem since - according to his own words - he has decided to not publish anything without financial compensation because this would make him communist.
Concerning Born rule I dare to have a different opinion than Lubos. I need not be afraid of Lubos's insults since Lubos as a brahmine of science refuses to comment anything written by inferior human beings like me and even refuses to mention their names: maybe Lubos is afraid of doing it might somehow infect him with the thoughts of casteless.
Without going to the details of quantum measurement theory, one can say that Born's rule is bilinear exression for the initial and final states of quantum mechanical transition amplitude. Bilinearity is certainly something deep and I will go to that below. Certainly Born's rule gives the most natural expression for the transition amplitude: demonstraing this is of course not a derivation for it.
- One could invent for the transition amplitude formal expressions non-linear in normalized initial and final states. One can however argue that the acceptable expressions must be symmetric in initial and final states.
- The condition that the transition amplitude conserves quantum numbers associated with symmetries suggests strongly that the transition amplitude is a function of the bilinear transition amplitude between initial and final states and norms of initial and final states. The standard form for non-normalized states - inner product divided by product of square roots norms - is indeed of this form. For instance, one could add exponentials of the norms of initial and final state norms.
- Projective invariance of the transition amplitude - the independence of the transition probabilities from normalization- implies that standard transition amplitude multiplied by a function - say exponential - of the modules square of the standard amplitude (transition probability in standard approach) remains to be considered.
- One could however still consider the possibility that the probability as given by Born rule are replaced by its function: pij→ f(pij)pij. Unitary poses strong constraints on f and my guess that f =1 is the only possibility.
Consider now the situation in TGD framework.
- In TGD framework the configuration space, "World of Classical Worlds" consisting of pairs of 3-surfaces at opposite boundaries of causal diamonds (CDs), is infinite-dimensional, and this sharply distinguishes TGD based quantum theory from wave mechanics. More technically, hyperfinite factors of type II (and possibly also III) replace factors of type I, in the mathematical formulation of the theory.
Finite measurement resolution is unavoidable and is represented elegantly in terms of inclusions of hyper-finite factors. This means that single ray of state space is replaced with infinite-D sub-space whose states cannot be distinguished from each other in given measurement resolution. The infinite-dimensional character of WCW makes the definition of the inner product for WCW spinor fields extremely delicate.
Note that WCW spinor fields are formally classical at WCW level and state function reduction remains the only genuinely quantal aspect of TGD. At space-time level one must perform second quantization of induced spinor fields to build WCW gamma matrices in terms of fermionic oscillator operators.
- WCW spinors which are fermionic Fock states associated with a given 3-surface. There are good reasons to believe that the generalization of the usual bilinear inner product defined by integration of the spinor bilinear over space (with Euclidian signature) generalizes but under extremely restrictive condition. Spinor bilinear is replaced with fermionic Fock space inner product and this bilinear is integrated over WCW.
The integration over WCW makes sense only if WCW allows a metric which is invariant under maximal group of isometries- this fixes WCW and physics highly uniquely. To avoid divergences one must also assume that Ricci scalar vanishes and empty space Einstein equations hold true. Metric determinant is ill-defined and must be cancelled by the Gaussian determinant coming from the exponent of vacuum functional, which is exponent of Kähler action if WCW metric is Kähler as required by the geometrization of hermitian conjugation which is basic operation in quantum theory. One could however still consider the possibility that the probabilities given by Born rule are replaced by their functions: pij→ f(pij)pij but unitarity excludes this. Infinite-dimensionality is thus quite not enough: something more is needed unless one assumes unitarity.
- Zero Energy Ontology brings in the needed further input. In ZEO the transition amplitudes correspond to time-like entanglement coefficients of positive and negative energy parts of zero energy states located at the opposite light-like boundaries of causal diamond. The deep principle is that zero energy states code for the laws of physics as expressed byS-matrix and its generalizations in ZEO
This implies that transition amplitudes is automatically bilinear with respect to positive and negative energy parts of zero energy state, which correspond to initial and final states in positive energy ontology. The question why just Born rule disappears in ZEO.
That ZEO gives justification also for Born rule is nice since it has produced a solution also to many other fundamental problems of quantum theory. Consider only the basic problem of quantum measurement theory due to determinism of Schrödinger equation contra non-determinism of state function reduction: Bohr's solution was to give up entirely ontology and take QM as a mere toolbox of calculational rules.
There is also the problem the relationship between geometric time and experienced time, which ZEO allows to solve and leads to much more detailed view about what happens in state function reduction. The most profound consequences are at the level of consciousness theory which is essentially generalization of ordinary quantum measurement theory in order to make observer part of the system by introducing the notion of self. ZEO also makes the physical theories testable: any quantum state can be achieved from vacuum in ZEO whereas in standard positive energy ontology conservation laws make this impossible so that at the level of principle the testing of the theory becomes impossible without additional assumptions.