### Low Energy Quantum Gravity and Matter-Gravity Entanglement

These were the topics Bernard Kay, from the University of York, talked to us about in a seminar last Friday. He has a rather ambitious theory, claiming to resolve in one single sweep many foundational issues in physics: the nature of black hole entropy, the information loss paradox, and the mysteries surrounding Schrodinger's Cat in ordinary quantum mechanics.

The theory has 5 axioms:

1) A quantum state is always represented by a pure state density operator.

2) Time evolution is unitary.

3) There is a well-defined split of the Hilbert space between a matter and a gravity component: H = Hm x Hg (please imagine that “x” enclosed in a circle).

4) Gravity degrees of freedom are unobservable. Observables on H take the form A x I where A is an observable on Hm and I is the identity on Hg.

5) The entropy of a black hole is the entropy of matter-gravity entanglement: the Von Neumann entropy of Hm when Hg is traced over (which is equal to that of Hg when Hm is traced over).

Bernard said these axioms are proposed for low energy quantum gravity, by which I understand he meant the ranges where quantum field theory in curved spacetime is normally used (e.g., black holes masses much larger than the Planck mass) though a calculation he did later used an even weaker, Newtonian approximation for gravity, which is also prominent in the paper the talk was based on.

According to Bernard, Axiom 4 explains why there are no macroscopic "Schrodinger Cat" quantum superpostions. Write such a superposition as a sum of two macoscopic states of matter M1, M2 with associated gravitational fields G1, G2. On the assumption of axiom 4, calculate the density operator for the matter subspace only; the link paper contains the calculation with a toy Newtonian model, which gives as result that the states M1, M2 appear decohered. The interference between them is supressed exponentially in the mass of the system divided by the Planck mass.

(When this point in the exposition was reached, Kirill Krasnov raised his hand to say: "But the gravitational field can be measured –by sitting in this room and not floating around I am measuring it!" Bernard asked him to be patient, but I’m not sure he provided an answer to this at the end. More later.)

On the other hand, it is Axiom 5 which provides a solution to the problems related with black hole entropy and evaporation. Bernard mentioned two such problems: The first is that usually in physics entropy is not a fully "objective" quantity because it is associated with a particular "coarse-grained" macroscopic description of a system; however, there seems to be nothing subjective in black hole entropy equaling A/4. The second is that if entropy increases during collapse to a black hole and subsequent evaporation, then evolution cannot be unitary because the Von Neumann entropy of a pure state is conserved in unitary evolution. The resolution offered to these problems is that the entropy is not that of the whole state but just that of the matter component of it, which grows because we ignore the (according to the axiom, unobservable) gravity part. Moreover, this generalises to arbitrary systems and even the whole universe, providing -according to Bernard- an explanation of the Second Law of Thermodynamics!

One argument Bernard used to support his position is that in standard accounts a black hole is in thermal equilibrium with its enviroment, and is thus described by a Gibbs mixed state which is the tensor product of a mixed state for matter and another one for gravity. Then the total entropy S of the system should be the sum of the entropies Sm and Sg. Which of this three entropies is equal to A/4? The "thermodynamical" calculation of black hole entropy points to Sm as the answer, while Hawking's derivation from the action of pure gravity points to Sg. In Bernard's picture, the total state is pure and its entropy is not the physical one measured; the partial entropies Sm and Sg turn out to be equal from axiom 5.

Going back to the interpretation of quantum mechanics, readers familiar with Penrose's views will notice a resemblance. But Kay's theory, as I understand it, is philosophically different from Penrose's. Penorse is worried by the

Some doubts and questions this theory leaves me with: The question about standard QM it purports to answer, "why don't we see macroscopic superpositions?", seems to me answered already by conventional decoherence effects; the ongoing mystery unsolved by that is what counts as a measurement and what is the ontological status of "wavefunction collapse", but Kay's theory seems silent on these issues. There is also the

The theory has 5 axioms:

1) A quantum state is always represented by a pure state density operator.

2) Time evolution is unitary.

3) There is a well-defined split of the Hilbert space between a matter and a gravity component: H = Hm x Hg (please imagine that “x” enclosed in a circle).

4) Gravity degrees of freedom are unobservable. Observables on H take the form A x I where A is an observable on Hm and I is the identity on Hg.

5) The entropy of a black hole is the entropy of matter-gravity entanglement: the Von Neumann entropy of Hm when Hg is traced over (which is equal to that of Hg when Hm is traced over).

Bernard said these axioms are proposed for low energy quantum gravity, by which I understand he meant the ranges where quantum field theory in curved spacetime is normally used (e.g., black holes masses much larger than the Planck mass) though a calculation he did later used an even weaker, Newtonian approximation for gravity, which is also prominent in the paper the talk was based on.

According to Bernard, Axiom 4 explains why there are no macroscopic "Schrodinger Cat" quantum superpostions. Write such a superposition as a sum of two macoscopic states of matter M1, M2 with associated gravitational fields G1, G2. On the assumption of axiom 4, calculate the density operator for the matter subspace only; the link paper contains the calculation with a toy Newtonian model, which gives as result that the states M1, M2 appear decohered. The interference between them is supressed exponentially in the mass of the system divided by the Planck mass.

(When this point in the exposition was reached, Kirill Krasnov raised his hand to say: "But the gravitational field can be measured –by sitting in this room and not floating around I am measuring it!" Bernard asked him to be patient, but I’m not sure he provided an answer to this at the end. More later.)

On the other hand, it is Axiom 5 which provides a solution to the problems related with black hole entropy and evaporation. Bernard mentioned two such problems: The first is that usually in physics entropy is not a fully "objective" quantity because it is associated with a particular "coarse-grained" macroscopic description of a system; however, there seems to be nothing subjective in black hole entropy equaling A/4. The second is that if entropy increases during collapse to a black hole and subsequent evaporation, then evolution cannot be unitary because the Von Neumann entropy of a pure state is conserved in unitary evolution. The resolution offered to these problems is that the entropy is not that of the whole state but just that of the matter component of it, which grows because we ignore the (according to the axiom, unobservable) gravity part. Moreover, this generalises to arbitrary systems and even the whole universe, providing -according to Bernard- an explanation of the Second Law of Thermodynamics!

One argument Bernard used to support his position is that in standard accounts a black hole is in thermal equilibrium with its enviroment, and is thus described by a Gibbs mixed state which is the tensor product of a mixed state for matter and another one for gravity. Then the total entropy S of the system should be the sum of the entropies Sm and Sg. Which of this three entropies is equal to A/4? The "thermodynamical" calculation of black hole entropy points to Sm as the answer, while Hawking's derivation from the action of pure gravity points to Sg. In Bernard's picture, the total state is pure and its entropy is not the physical one measured; the partial entropies Sm and Sg turn out to be equal from axiom 5.

Going back to the interpretation of quantum mechanics, readers familiar with Penrose's views will notice a resemblance. But Kay's theory, as I understand it, is philosophically different from Penrose's. Penorse is worried by the

*ontological*interpretation of a superposed state, and posits an objective collapse of the wavefunction driven by quantum gravity. Kay does not seem to address the ontological problem: his solution is very similar to the standard "fapp" solution based on decoherence, with the difference that instead of an "enviroment" Kay has "the gravitational field" as the unobserved degrees of freedom that make interference between macroscopic lumps of matter supressed. Towards the end of the talk Kay mentioned that his theory makes predictions undistinguishable from those of standard quantum mechanics, in particular for the crucial experiment to test macroscopic superpostions that Penrose has proposed (described in his*The Road to Reality*). I asked after the talk how could predictions from this theory be identical to standard QM if macroscopic superpositions are unobservable in principle in it; in standard QM macroscopic superpositions are only unobservable in practice, because of enviromental decoherence, but with enough care protecting a system from decoherence they can be observed regardless of the mass. He said he needed to think further on that question.Some doubts and questions this theory leaves me with: The question about standard QM it purports to answer, "why don't we see macroscopic superpositions?", seems to me answered already by conventional decoherence effects; the ongoing mystery unsolved by that is what counts as a measurement and what is the ontological status of "wavefunction collapse", but Kay's theory seems silent on these issues. There is also the

*ad hoc*assumption that gravity is unobservable, and the need to answer Kirill's question (if our measuements of gravity are said to be "indirect", why aren't those of other fields indirect in the same way?) And black hole entropy seems much more likely to be related to correlations between the observed exterior and the unseen interior, than to correlations between matter and gravity as such, where I see no*a priori*reason to declare one of them unobservable. Other opinions?
## 1 Comments:

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By sports handicapping software, at 2:00 AM, April 28, 2012

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