Let me begin my talkFootnote 1 by recalling one version of the second law of thermodynamics:
The entropy of the universe begins low and increases monotonically.
There are long-established and well-known arguments—see the discussion of ‘branch systems’ in [1] as also reviewed e.g. in [2])—that other statements of the second law, in terms of what can and cannot happen with heat engines, refrigerators etc. follow from the above statement. As also explained in these references, the above statement leads to an explanation of time asymmetry; i.e. why, for example, it is commonplace to observe wine-glasses fall off tables and smash into pieces, but we never see lots of smashed pieces assemble themselves into wine-glasses and jump onto tables (Fig. 1).
But how do we define the entropy of a closed system? And why does it increase?
A standard way of answering this (essentially due to Boltzmann around 1870) might be to consider for example what will happen if one starts with a system of N gas molecules in the left half of a box (see Fig. 2) and removes a partition, allowing the particles to diffuse into the right half of the box.
In a classical discussion, one describes the states of this system with some given energy in terms of a \(6N-1\) dimensional phase space, the points of which are called ‘microstates’ and (see Fig. 3) one imagines this phase space to be divided up into cells—called ‘macrostates’—with the property that we cannot in practice distinguish between any pair of microstates in any single macrostate. One then defines the (‘coarse-grained’) entropy, S, of a microstate by
$$\begin{aligned} S = k\log W \end{aligned}$$
(1)
where k is Boltzmann’s constant and W is the volume of the macrostate containing that given microstate.
The standard argument then is that (see Fig. 3) the macrostate corresponding to “all the particles are in the left half of the box” will have a vastly smaller volume in phase space than the large macrostate which corresponds to “the molecules fill the box with roughly uniform density”. Hence, as time goes on and the state of the system wanders around the phase space accordingly, it is highly likely that the entropy—as defined by (1) will get bigger and stay bigger.
However, this definition of entropy and this argument for its increase depends, unsatisfactorily, on the need to make judgments about what we can distinguish. For example, if (see again Fig. 3) after previously ignoring such fine distinctions, we were to take the view that we can distinguish a state where, say, 48% of the particles are in the left half of the box and 52% in the right half from a state with roughly equal proportionsFootnote 2 then, at times for which the system’s microstate lies in the accordingly-defined new macrostate (obviously a subregion of the previously discussed large macrostate) then Eq. (1) would ascribe a different value to the entropy.
Moreover, this unsatisfactory arbitrariness and vagueness in the definition of entropy is even more of a problem if we want to account for the version of the second law with which we began. For we are not even present to make any distinctions in the early universe!
Turning to the quantum setting, von Neumann gave us long ago a quantum translation of Boltzmann’s equation (1). Given a description of our system in terms of a density operator, \(\rho \) acting on the system’s Hilbert space \(\mathcal {H}\), one defines its von Neumann entropy, \(S^{\mathrm {vN}}(\rho )\), by
$$\begin{aligned} S^{\mathrm {vN}}(\rho )=-k\mathrm{tr}(\rho \log \rho ). \end{aligned}$$
(2)
But if we were to equate the physical entropy, \(S^{\mathrm {physical}}\), with \(S^{\mathrm {vN}}(\rho )\) and if \(\rho \) satisfies the usual unitary time evolution rule
$$\begin{aligned} \rho (t)=U(t)\rho (0)U(t)^{-1} \end{aligned}$$
then we would conclude that
$$\begin{aligned} S^{\mathrm {physical}}(\rho (t))= {\mathrm {constant}}. \end{aligned}$$
in contradiction with the second law. We shall call this the second law puzzle. One can overcome this difficulty by defining quantum counterparts to the above classical coarse-graining, but of course one then would have the same unsatisfactory vagueness and subjectivity as we discussed above in the classical case.
More interestingly, one can seek to exploit a feature of quantum mechanics which has no classical counterpart: If we have a pure state, described by a density operator, \(\rho =|\varPsi \rangle \langle \varPsi |\), which is a projector onto a vector, \(\varPsi \), in a Hilbert space, \(\mathcal {H}_{\mathrm {total}}\), which arises as the tensor product,
$$\begin{aligned} \mathcal {H}_{\mathrm {total}}=\mathcal {H}_\mathrm {A}\otimes \mathcal {H}_\mathrm {B} \end{aligned}$$
of two Hilbert spaces, \(\mathcal {H}_\mathrm {A}\) and \(\mathcal {H}_\mathrm {B}\), then the reduced density operator, \(\rho _A\) on \(\mathcal {H}_\mathrm {A}\), defined as the partial trace, \(\mathrm{tr}_{\mathcal {H}_\mathrm {B}}(\rho )\), of \(\rho \) over \(\mathcal {H}_\mathrm {B}\), will typically have \(S^{\mathrm {vN}}(\rho _\mathrm {A}) \!>\! 0\).
We remark that
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This partial trace is characterized by the property that, if O is a (self-adjoint) operator on \(\mathcal {H}_\mathrm {A}\), then
$$\begin{aligned} \mathrm{tr}(\rho _\mathrm {A} O)_{\mathcal {H}_\mathrm {A}}= \langle \varPsi (O\otimes I)|\varPsi \rangle _{\mathcal {H}_{\mathrm {total}}}. \end{aligned}$$
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Both reduced density operators have equal von Neumann entropies:
$$\begin{aligned} S^{\mathrm {vN}}(\rho _A)=S^{\mathrm {vN}}(\rho _\mathrm {B}) \end{aligned}$$
(3)
and this common value is often known as the A–B entanglement entropy of the total state-vector \(\varPsi \).
In a variant of the ‘environment paradigm for decoherence’ or, from another point of view, a variant of a possible approach to quantum statistical mechanics, this formalism is often applied in the case that A is interpreted as standing for some ‘system’ and B for the system’s ‘environment’ or ‘energy bath’ and \(S^{\mathrm {vN}}(\rho _\mathrm {A})\) is then interpreted as the entropy of the system due to entanglement with the environment.
So the environment paradigm gives us an objective notion of entropy. However, there remain problems:
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It only offers a notion of entropy for open systems.
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There are lots of ways of decomposing a given \(\mathcal H\) as \(\mathcal {H}_\mathrm {A}\otimes \mathcal {H}_\mathrm {B}\). How we choose to decompose it depends on subjective choices and, again, we are not around in the early universe to make those choices.
What I’d like to point out is that one can envisage an alternative physical use of this mathematical fact: Suppose there’s some decomposition that’s physically natural, then maybe we could define the entropy of a total closed system by
$$\begin{aligned} S^\mathrm {total}= S^{\mathrm {vN}}(\rho _\mathrm {A}) \quad (= S^{\mathrm {vN}}(\rho _\mathrm {B})) \quad \hbox {(= A--B entanglement entropy)} \end{aligned}$$
(4)
rather than interpreting this mathematical quantity as the entropy of the A-subsystem!
We propose that the identification:
$$\begin{aligned} \hbox {A}=\textit{matter}; \quad \hbox {B}=\textit{gravity}, \end{aligned}$$
is the right choice. This is our matter-gravity entanglement hypothesis. (See [3,4,5] for early papers, and [6] and the remainder of the present article for recent partial overviews and further references.)
In support of this, we note that the decomposition has to be meaningful throughout the entire history of the universe: E.g. we could not identify A with photons and B with nuclei + electrons because these notions are not even meaningful until the photon epoch. We content ourselves, though, with going back to just after the Planck epoch; we assume that a low-energy quantum gravity theory holds there and throughout the entire subsequent history of the universe and that this is a conventional (unitary) quantum theory with \(\mathcal {H}=\mathcal {H}_{matter}\otimes \mathcal {H}_{gravity}\). We will also assume that the initial degree of matter-gravity entanglement is low. (We leave it for a future theory of the pre-Planck era to explain that.)
These assumptions then appear to be capable of offering an explanation of the second law in the form stated at the outset since one can argue that an initial state with a low degree of matter-gravity entanglement will, because of matter-gravity interaction, get more entangled, plausibly monotonically, as time increases. At least the question of whether the second law holds becomes a question which, in principle, can be answered mathematically once we specify the (low-energy) quantum gravity Hamiltonian (i.e. the generator of the unitary time-evolution) and the initial state. What we have called the second law puzzle would then be resolved because once we define entropy as matter-gravity entanglement entropy (rather than as the von Neumann entropy of the total state) there is no conflict between its increase and a unitary time-evolution.