## Saturday, April 08, 2006

### Timeless physics

Nima Arkani-Hamed is organizing conceptual seminars on Friday dedicated primarily to quantum cosmology. Today, he defined the topic as timeless quantum physics. Previously, he had explained the minisuperspace approximation, the Wheeler-DeWitt equation, the no-boundary wavefunction, decoherence, and many other standard but not generally known insights very nicely, making most of the audience rather excited.

But his main goal is to create something new and ambitious, namely a framework for quantum physics that does not require any time. The readers of Chapter 15 of Brian Greene's "The Elegant Universe" know very well what I mean and why it is so hard. Many of us have been thinking about these issues for hundreds of hours.

To achieve his dreams, Nima considers a timeless wavefunction in an extended Hilbert space in which time becomes a new observable much like "x" (which itself seems appropriate for mechanics only and fails in the context of relativistic field theory, but there are many simpler difficulties so let's not go into these advanced ones). In the case of one-dimensional quantum mechanics, his wavefunction can be constructed from the usual solution of the Schrödinger equation via the formula
• psi(t,x) = integral dt /t> (x) /psi(t)>
and a special constraint derived in the obvious way replaces the usual Schrödinger equation. The inner product can be defined with an extra integral over "dt", but you must divide the result by "delta(0)" - something that I believe only makes sense if the picture is actually equivalent to the usual Hilbert space. More generally, I understand this change of notation when we construct something completely equivalent to the usual quantum mechanics with time but so far I have failed to understand all the hypothetical new ideas.

When one sacrifices time and treats it as emergent, it makes the definition of unitarity more subtle. Nima obviously wants to relax the strict rules of unitarity - so that universes can be created and destroyed - but I don't understand what are the new rules that should replace it. Surely we don't want to relax the requirement of unitarity completely because what we would obtain are rules that allow completely non-unitary - and therefore physically meaningless - theories.

Alternatively, if we had modified the rules to calculate the probabilities in such a way that the total probability would equal one, we would introduce rather generic non-local interactions or, even more seriously, the Everett phone - a tool to communicate with the other parallel universes in the Many Worlds interpretation of quantum mechanics, as I was explained. I view all these effects as lethal inconsistencies.

We obviously only want to allow very mild violations of unitarity around the Big Bang but what the word "mild" means is, so far, beyond me. We could decide to tolerate a "small" non-unitarity or other "small" inconsistencies but I wouldn't treat such an approach as rules of new meaningful physics: indeed, everything we added to the previous framework is rubbish. The amount of new physics is exactly equal to the amount of inconsistency we added. It's just like if you study a non-renormalizable theory above the scale where it breaks down.

Also, one of Nima's other ambitious statements is that different cosmologies based on different points of the landscape are just different ways to identify time-slicing within the same timeless Hilbert space; the wavefunction is identical for all conceivable evolutions.

I've tried to understand what these things are supposed to mean but so far I failed. Surely, all infinite-dimensional separable Hilbert spaces are unitarily equivalent as a famous theorem informs us. If you take the Hilbert space of a harmonic oscillator, you can also use it as the Hilbert space for the Standard Model as long as you define a Standard Model Hamiltonian (and other observables) on it. The Standard Model Hamiltonian and the harmonic oscillator Hamiltonian won't commute with each other - much like two generic operators that have no relation with each other. In such cases, if you use completely different Hamiltonians (generators of time evolution) for the same Hilbert space, I think you should have really used two different Hilbert spaces.

Nima was explaining me that there is something nontrivial about the picture of the "unified state for all backgrounds" once you include the assumption that all observers live for a finite time, but again, it is probably too difficult for me to comprehend. Moreover, I think that in infinite space, we explicitly know that two different vacua on the moduli space are different, not identical, states in the full Hilbert space - they are the ground states of different superselection sectors. This intuition based on superselection sectors could be circumvented in quantum cosmology but it would be nice to see a real argument and new rules that replace the old ones.

I remain very conservative about all these questions. Every working quantum theory must have different moments and exactly unitary operators that tell us how to evolve from one moment to another. In the context of quantum gravity, you may want to study the S-matrix only because it's the only "nice" gauge-invariant observable. Even if you choose to focus on the S-matrix, you have the moment "before" at minus infinity and "after" at plus infinity and the S-matrix must be unitary.

Moreover, you can probably always choose some gauge-fixing of the diff symmetry if you don't care about a manifest Lorentz symmetry - for example you may pick a gauge generalizing the light-cone gauge - and in these variables, quantum gravity behaves just like any other quantum theory that depends on time. Indeed, the gauge-fixing procedure may have subtleties when your spacetime has horizons or other topologically nontrivial structures, but these problems should be investigated separately: there are physical situations in which they don't appear at all.

In this sense, I believe that an additional discussion of the horizons or singularities as "pointlike anomalies" should be the only conceivable addition of quantum gravity to the usual basic postulates of quantum mechanics that all physicists learn in the college. Of course, I agree with Nima that there could be larger Hilbert spaces constrained by identifications - but in my opinion, all these manipulations only affect the formalism, the description with which we want to obtain the final result: they don't give us any new rules that could be used to solve the vacuum selection problem and similar puzzles.

The different definitions of time on the Hilbert space are also relevant for the hypothetical black hole complementarity principle according to which the interior and the exterior of the black hole recycle the same Hilbert space - but I feel that I can't write anything new about these things.

Timeless physics is an extremely provoking and intriguing possibility that intuitively seems relevant for quantum cosmology. Whenever one starts to think about these issues, it seems that all rules (and dynamical rules in particular) start to evaporate or they at least get diluted. Someone else - like a Massachusetts lawyer in the audience - may enjoy such an evaporation but I personally don't. If you don't replace the diluted rules by rules that are exactly as strong as the rules in the familiar context - or, more optimistically, by even stronger rules - I will probably always view such speculations as negative progress.

#### 1 comment:

1. It is an absolutely correct direction to study timeless physics, i.e., consider time as emergent, not fundamental. Actually one should go one step further, not only is the universe time-less, it is also space-less. Both space and time are emergent, and not fundamental. The only thing fundamental is quantum entropy, which is conserved. On top of that you construct spacetime. And all physics phenomena we observe are simply geometries on the emergent spacetime.

Unitarity is important and of course it contradicts the creation and destruction of the universe. Nima was wrong in trying to relax unitarity to reconcile that. The correct answer is there is no creation and detruction of the universe. Something time-less can neither be created or destructed. It's like a DVD movie. It gives you an illusion of time and events that's happening but it is actually just one static object. Once you finish playing the DVD, you put it back into the jewel box. Nothing is created and nothing is destroyed, and the movie you just saw was merely some emergent phenomena, not something real.

Quantoken