# Wheeler-DeWitt and timelessness

1. Nov 3, 2014

I am missing something in my reading of the article "Time from quantum entanglement: an experimental illustration" (http://arxiv.org/abs/1310.4691). One of its premises is that Wheeler-DeWitt equation makes the universe appear as "timeless" from the "outside" (whatever that it: the same problem in positing a universal wave function, but let's not get into that). But in reading descriptions of this (e.g., http://en.wikipedia.org/wiki/Wheeler–DeWitt_equation) although time is not explicitly mentioned in the equation, it is implicit in the way it is set up. So can one really talk about a timeless universe?

Then, another problem: the "Time from quantum entanglement" paper purports to present time as an emergent phenomenon from entanglement, but the toy universe presented there seems to use time in its set-up, so that it seems one is defining time in terms of time. So is time really an emergent phenomenon there?

Oh, and since the Wheeler-deWitt equation has both quantum and general relativistic elements, I wasn't sure which of the two rubriks to put it under.

Thanks in advance for setting me straight.

2. Nov 4, 2014

### atyy

The timelessness of the Wheeler-DeWitt equation is not simply the same as that arising from a universal wave function. The Wheeler-DeWitt equation comes from the attempt to canonically quantize gravity. Because of the gauge-redundancy of gravity, "time evolution" takes one state into the same state.

3. Nov 4, 2014

### Demystifier

Time evolution in various interpretations of QM

Consider a closed system in a state with definite energy. (A Universe described by Wheeler-DeWitt equation is just a special case, in which the definite energy turns out to be zero.) How time evolution is possible in such a system? Unfortunately, an interpretation-independent answer to this question does not exist. Here I briefly review different answers provided by different interpretations.

Copenhagen''-collapse interpretation
According to this interpretation proposed by von Neumann, everything, including the observer, is described by the wave function. However, the time-evolution of the wave function is not always governed by the Schrodinger equation. Instead, the act of observation is associated with a wave-function collapse. The collapse introduces an additional time-evolution in the system, not present in the evolution by the Schrodinger equation. In this interpretation the act of observation plays a fundamental role, but the concept of observation itself is not described by physics.

Copenhagen'' interpretation with classical macro-world
According to this interpretation, usually attributed to Bohr, quantum mechanics can be applied only to the micro-world, not to the macro-world. The macro-world is described by classical mechanics, so the time evolution in the macro-world is not governed by a Schrodinger equation. In a closed system a quantum micro-subsystem interacts with a classical macro-subsystem, so that the time-dependence of the latter induces a time dependence of the former.

Modern instrumental Copenhagen'' interpretation
This is a widely-used practically oriented interpretation of QM (see e.g. the book by Peres), in which QM is nothing but a tool used to predict the probabilities of measurement outcomes for given measurement preparations. The measurement preparations are freely chosen by experimentalists. The experimentalists themselves are not described by QM. The free manipulations by experimentalists introduce additional time-dependence in the system not described by the Schrodinger equation. Within such an interpretation, the concept of wave function of the whole Universe does not make sense.

Objective collapse
In this interpretation the Schrodinger equation is modified by adding a stochastic term due to which the wave function collapses independently on any observers. The best known example of such a modification is the GRW theory.

Hidden variables
In this class of interpretations, the physical objects observed in experiments are not the wave functions, but some other time-dependent variables $\lambda(t)$. Even if the wave function governed by the Schrodinger equation is time-independent, the hidden'' variable $\lambda(t)$ may depend on time. The best known and most successful model of such variables is given by the Bohmian interpretation.

Statistical ensemble
According to this interpretation, the wave function is only a property of a statistical ensemble of similarly prepared systems and tells nothing about properties of individual physical systems. So if a wave function is time-independent, it does not mean that individual systems do not depend on time. This interpretation can be thought of as an agnostic variant of the hidden-variable interpretation, in the sense that the existence of hidden variables is compatible (and perhaps even natural) with the statistical-ensemble interpretation, but the statistical-ensemble interpretation refrains from saying anything more specific about them.

Consistent histories
In this interpretation, the wave function is a tool to assign a probability to a given time-dependent history of the physical system. In this sense, it is similar to hidden-variable interpretations. However, to avoid non-localities typically associated with normal hidden-variable theories, the consistent-histories interpretation replaces the classical propositional logic with a different kind of logic.

Many worlds
According to the many-world interpretation, the Universe as a whole is nothing but a wave function evolving according to the Schrodinger equation. So, if wave function of the Universe is a state with definite total energy, at first sight it seems impossible to have any nontrivial time-dependence in the system. Nevertheless, a non-trivial time dependence can be introduced in a rather subtle way, by redefining the concept of time itself. Even if $\psi(x_1,\ldots,x_N)$ does not depend on an evolution parameter $t$, some of the configuration variables $x_1,\ldots,x_N$ may represent readings of a physical clock, on which $\psi(x_1,\ldots,x_N)$ still depends. In such an interpretation of QM all probabilities are interpreted as conditional probabilities.

Of course, none of these interpretations is without difficulties. However, to avoid controversy and keep neutrality, the difficulties will not be discussed. I expect that critical readers will immediately recognize some difficulties with most of these interpretations, even without my assistance.

Last edited: Nov 4, 2014
4. Nov 4, 2014

### Demystifier

Now, after the brief review above, let me comment the paper mentioned in the first post. It seems to me that the authors assume a variant of the last (i.e. many-world) interpretation on my list.

5. Nov 4, 2014

Thanks for the replies, atty and Demystifier.

atty: so would you say that for every invariant quantity Q in a gauge theory, that the space described by the theory is Q-less?

Demystifier: so if the paper in question is adopting the last-named interpretation in your summary, then it sounds as if you would agree that time might be implicit in the Wheeler-deWitt equation, although it does not explicitly appear?

6. Nov 4, 2014

### atyy

I don't know what you mean. The extent to which the WdW equation is "timeless" is discussed in http://arxiv.org/abs/0812.0177

"For simple systems, including Minkowskian field theories, the Hamiltonian formulation generally serves as the royal road to quantum theory. It was therefore adopted for quantum gravity by Dirac, Bergmann, Wheeler and others. But absence of a background metric implies that the Hamiltonian dynamics is generated by constraints [17]. In the quantum theory, physical states are solutions to quantum constraints. All of physics, including the dynamical content of the theory, has to be extracted from these solutions. But there is no external time to phrase questions about evolution."

"Let us begin with the issue of time. In the classical theory, one considers one space-time at a time and although the metric of that space-time is dynamical, it enables one to introduce time coordinates —such as the proper time– that have direct physical significance. However in the quantum theory — and indeed already in the phase space framework that serves as the stepping stone to quantum theory — we have to consider all possible homogeneous, isotropic space-times. In this setting one can introduce a natural foliation of the 4-manifold, each leaf of which serves as the ‘home’ to a spatially homogeneous 3-geometry. However, unlike in non-gravitational theories, there is no preferred physical time variable to define evolution. As discussed in section II, a natural strategy is to use part of the system as an ‘internal’ clock with respect to which the rest of the system evolves. This leads one to Leibnitz’s relational time."

"It turns out that the WDW theory leads to similar predictions in both k=0 and k=1 cases [20, 23, 27, 28]. They pass the infra-red tests with flying colors (see Fig 2). But unfortunately the state follows the classical trajectory into the big bang (and in the k=1 case also the big crunch) singularity. Thus the first of the possibilities listed above is realized. The singularity is not resolved because expectation values of density and curvature continue
to diverge in epochs when their classical counterparts do."

7. Nov 4, 2014

### atyy

It looks like there is indeed "external time" in the universe in their set-up. I think the reason they can say that there is no time evolution is that the system is in a stationary state (eigenstate of the Hamiltonian), which only evolves by an overall change of phase with respect to the external time, which means that all probabilities seen by an external observer do not change with respect to external time.

8. Nov 4, 2014

### bhobba

And there are others as well - although that list is pretty complete. Every single one has difficulties - every single one - even mine. But the interesting thing is pick an issue, not just time evolution mentioned here, but many others, and you will find interpretations with different takes and difficulties. And of course what one person considers a difficulty another couldn't care less about - which makes this whole interpretation business a horrid minefield.

The way out of course is to experimentally decide between them, but until someone figures out how to do that - well the debate continues.

Thanks
Bill

Last edited: Nov 4, 2014
9. Nov 4, 2014

Thank you for your replies, atyy and bhobba.

atyy: your answers were very informative, and not only was the excerpt out of the article enlightening, but also the rest of the article (which I am still working my way through) is extremely interesting. Thanks again.

bhobba: what is an example of "what one person considers a difficulty another couldn't care less about": aren't all difficulties one of two: internal and external consistencies? (i.e., lack of logical contradiction, and agreement with experiment?) Aren't both types of difficulties bothersome for anyone?

10. Nov 5, 2014

### bhobba

Take Bohmian Mechanics (BM).

Its issue is the inherent un-observability of its pilot wave. The central lesson of SR is you don't introduce inherently unobservable things like an aether just to retain your intuition of how the world works. For me that's a deal breaker - but for others they couldn't care less - what it resolves for them is far more important.

Others don't like the inherent non-locality of BM either, or the fact it reintroduces a preferred frame - personally it doesn't worry me - but for some a big no no. Interestingly I know at least one guy for whom that's a big plus:
http://ilja-schmelzer.de/glet/

Thanks
Bill

11. Nov 5, 2014

### Demystifier

12. Nov 5, 2014