Real-time gauge/gravity duality

Posted by Balt van Rees

May 25, 2009

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Balt van Rees from the U. of Amsterdam continues the discussion of non-equilibrium AdS/CFT we have started not so long time ago. Since his recent paper with Skenderis was one of the major achievements in the field, I highly recommend going through his post. Dmitry.

Our recent paper Real-time gauge/gravity duality offers a prescription for the computation of real-time correlation functions using the gauge/gravity duality (i.e. the AdS/CFT correspondence and its generalizations). In this post I would like to explain the motivation for the prescription as well as some general ideas; technical details and additional references can be found in the paper.

Let us begin by considering quantum field theory in Lorentzian signature spacetimes. Such ‘real-time’ quantum field theory is somewhat more complicated than its corresponding Wick-rotated counterpart: one finds lightcone singularities, $i\epsilon$ insertions, operator orderings, non-uniqueness of classical propagators and the need to carefully specify initial and final state or density matrices. On the other hand, all this extra structure precisely allows for the dynamics in our everyday lives and we therefore better know how to deal with these real-time subtleties!

Fortunately, in quantum field theory the situation is at least conceptually well under control. Most of the complications of the previous paragraph are in fact related and can be brought together rather elegantly. Since we will need some aspects of real-time quantum field theory below, let us give some details on how this is done.

Real-time quantum field theory
Consider a real-time quantum field theory path integral. It allows us to compute time-ordered correlation functions, which have specific analyticity properties (often specified in terms of insertions of factors of $i \epsilon$ ). As we will now review, these analyticity properties are intimately related to the proper specification of the initial and final state in the path integral.

Let us for example suppose that the initial and final state are the vacuum state $|\Omega \rangle$ of the theory. In quantum mechanics the wave function of the vacuum $\langle x | \Omega \rangle$ can be obtained by computing

$\lim_{\beta \to \infty}\langle x | e^{- \beta H} | \Psi \rangle$ (1)

for some position eigenstate $|x\rangle$ and an arbitrary state $|\Psi\rangle$ . Indeed, inserting a complete set of energy eigenstates in (1) and taking the limit projects upon the vacuum state $|\Omega\rangle$ (which we take to have zero energy) and we find the vacuum wave function up to an overall factor $\langle \Omega | \Psi \rangle$ that does not depend on .

Since the factor $e^{-\beta H}$ in (1) is just the time evolution operator over an imaginary time interval of length $\beta$ , the amplitude (1) can be computed by a Euclidean path integral over a segment in imaginary time $\tau=i t$ of length $\beta$ . The above arguments also hold in quantum field theory, and we may conclude that vacuum-to-vacuum amplitudes like the familiar real-time partition function,
$\langle \Omega | \exp(- i \int J O) | \Omega \rangle$
are computed by path integrating not only along the real time axis with a source for an operator , but rather by adding two semi-infinite Euclidean segments on both the initial and final end of the Lorentzian segment as well. The complete contour is sketched in figure 1.

Figure 1. Contour in the complex time plane corresponding to vacuum amplitudes. The vertical (Euclidean) segments are extended to infinity, and the crosses indicate possible operator insertions.

Fig.1. Contour in the complex time plane corresponding to vacuum amplitudes. The vertical (Euclidean) segments are extended to infinity, and the crosses indicate possible operator insertions.

As we mentioned above, the ‘physical’ effect of these wave function insertions in a Lorentzian path integral is that they lead precisely to the familiar factors of $i\epsilon$ in correlation functions. A quick (but nonrigorous) way to see this relation is the following: consider a mode of the form $\exp (-i \omega t)$ in the Fourier transform of the two-point function $\langle T O(t,x) O(0,0) \rangle$ . At a certain large time we turn to the Euclidean segment and substitute $t=T – i \tau$ with $\tau > 0$ , so our mode becomes $\exp(- i \omega T – \omega \tau)$ . We now only find regularity as $\tau \to \infty$ if $\omega > 0$ : this is precisely compatible with Feynman’s $i\epsilon$ -prescription corresponding to ‘positive frequencies to the future’ (and similarly we obtain ‘negative frequencies to the past’). For a proper derivation of these facts, see for example Weinberg or the appendix of our paper.

Translation to gauge/gravity duality
Let us now turn to string theory and in particular the gauge/gravity duality. According to the standard dictionary, we should regard spacetimes as semiclassical approximations to some ’stringy path integral’ with boundary conditions for the fields specified by the field theory sources. However, in Lorentzian signature a quantum field theory partition function also depends on initial and final states (in the above example we have just seen that this dependence is nontrivial even for the vacuum state). Correspondingly, one expects that such states need to be specified in the bulk as well. So what would be the precise map between bulk and boundary states?

In our proposal (using an idea of Maldacena), we very literally translate the aforementioned Euclidean path integral approach to the bulk theory. We thus propose to take into account the entire contour for the field theory path integral and not just the Lorentzian segments. This means that we have to ‘fill in’ the entire contour with a (d+1)-dimensional asymptotically AdS spacetime, which then consists of various ’segments’: those that end on the Euclidean segments of the contour generally have a positive definite bulk metric, whereas those ending on the Lorentzian segments have a Lorentzian metric. These segments are then glued together along bulk hypersurfaces that should end precisely on the corners of the contour. We can for example `fill’ the contour of figure 1 (times a circle in d = 2), with segments of three-dimensional Euclidean and Lorentzian AdS, leading to the bulk space of figure 2 (where we actually compactified the semi-infinite Euclidean segments to hemispheres).

Figure 2. The bulk space(time) corresponding to the contour of figure 1, with the operators now inserted on the bundary of the bulk space.

Nicely enough, the construction fits in naturally with the Hartle-Hawking proposal for a ‘wave function of the universe’. The hypersurfaces where the metric jumps should then be seen as the moment where we begin and end a ‘time’ evolution in quantum gravity, and the Euclidean segments then provide exactly a wave function of metrics for these initial and final surfaces. The particular extension of the corners in the boundary contour to bulk hypersurfaces is actually irrelevant for physical observables, which makes the setup very ‘holographic’ as well.

We mentioned before that taking account of the initial and final states in a field theory path integral leads to the correct $i \epsilon$ insertions in correlation functions. On the other hand, in the gauge/gravity duality we compute these correlation functions as functional derivatives of an on-shell supergravity action with respect to its (radial) boundary data. Does our construction then indeed lead to the correct $i \epsilon$ insertions if we compute the correlation functions in this way? This indeed seems to be the case, since all examples we worked out agree with field theory expectations. The (again nonrigorous) explanation is that the Euclidean segments account for the proper $i \epsilon$ insertions in the bulk-boundary propagators, which then map one-to-one into the same $i \epsilon$ insertions in the boundary correlation functions.

Future directions
Let us finally mention some applications and directions for future research. First of all, the prescription allowed us to rederive and generalize the popular recipe by Son and Starinets (see also this paper by Herzog and Son) for thermal two-point functions obtained from black hole backgrounds. We expect further applications to arise in the study of nonequilibrium systems in the bulk and boundary theories, notably the collapse of matter to black holes and the associated thermalization in the boundary theory. The prescription may be particularly useful in studying the holographic encoding of global properties in the bulk, for example the presence of bulk horizons and the regions beyond the horizon. It would also be interesting to further extend the analogy with the Hartle-Hawking wave function and to study aspects of Euclidean versus Lorentzian string theory beyond the supergravity approximation.

AdS/CFT, Journal club, Master Mason, Quantum field theory, String theory

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