# I Problem of time in quantum field theory?

#### PrashantGokaraju

The usual manifestly unitary quantum theory can be described either in the schrodinger or heisenberg pictures. Both of these pictures are 'non-relativistic', because either the operators or the states evolve in time depending on the picture. The path integral formulation is manifestly relativistic, but in its present form is incomplete in the sense that unitarity is not manifest.

It is possible to interpret canonical quantum mechanics in a relativistic way if we consider the creation operators to be the 'string field theory' limit of quantum gravity.

a+ = x + ip creates a particle of momentum p. This state |p⟩ evolves in time. The corresponding relativistic notion would be a creation operator that creates a 'spacetime history', rather than a state that evolves in time. An example of a spacetime history is exp i(px - Et), which is characterized not by an x and a p, but by E and p. The creation operators satisfy

a+a- = N

where N is the number of particles. This means that in a sense a ~ √N. This property of the creation/annihiliation operators must be preserved by a covariant version of these operators for the following reason. The difference between time and space means that particles should be considered to be 'packets of energy', but there is no notion of a 'packet of momentum'. Momentum is the flux of energy. Covariant theories treat time and space on an 'equal footing', but this is very subtle for the reason mentioned.

For example, in loop quantum gravity, the 'Wheeler-de Witt equation'

H|ψ⟩ = 0

Has de Sitter spacetime as a solution (ψ = ei(dA^A + A^A^A).) Where the term in the exponent is the action for Chern-Simons theory. This equation presumably corresponds to

(p2 - m2)φ = 0

Where now p2 - m2 is in some sense a Hamiltonian, and φ is a spacetime history rather than a state that evolves in time.

Please feel free to ask any questions, as there are many details to include in one post.

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#### PrashantGokaraju

One more thing to mention:

1/p2 - m2 = ∫ exp τ(p2 - m2)

where the limits are 0 and ∞.

In this sense p2 - m2 is a Hamiltonian conjugate to the 'time' τ. (in string theory there is no difference between time and space.)

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#### Demystifier

2018 Award
in string theory there is no difference between time and space.
What is that supposed to mean?

#### PrashantGokaraju

I think it is related to the fact that at the plank scale, the Casimir forces due to vacuum fluctuations and the interactions by virtual exchanges or electrostatic forces become of the same order of magnitude, and you cannot really distinguish between them.

The energy levels of a harmonic oscillator are given by (n + 1/2)hω where n is the number of particles. The vacuum energy or ground state energy 1/2 hω is presumably not visible in minkowski space because it is flat, but probably there should be some way to see Minkowski space as de Sitter space with a cosmological constant Λ, because the minkowski vacuum appears to be excited in rindler or accelerated coordinates. The cosmological constant is in a sense a global concept.

#### weirdoguy

I think it is related to the fact that at the plank scale, the Casimir forces due to vacuum fluctuations and the interactions by virtual exchanges or electrostatic forces become of the same order of magnitude, and you cannot really distinguish between them.
Sorry, but I don't think that any of that makes sense at all.

The question was: what does it mean that "in string theory there is no difference between time and space".

#### PrashantGokaraju

The open strings and closed strings are in some sense equivalent because of the equivalence between gauge theory and gravity. since the boundary and the bulk are of different dimension, space dimensions in the bulk could be equivalent to time dimensions on the boundary.

#### Demystifier

2018 Award
The open strings and closed strings are in some sense equivalent because of the equivalence between gauge theory and gravity. since the boundary and the bulk are of different dimension, space dimensions in the bulk could be equivalent to time dimensions on the boundary.
No, the bold part is not what the gauge/gravity correspondence tells.

#### PrashantGokaraju

The diffeomorphism constraint equation mentioned before is about gravity in 4 dimensional spacetime, but its solution in terms of Chern-Simons theory is 3 dimensional. So the time dimension is somehow present in the 3 dimensional solution representing de Sitter space.

#### Auto-Didact

^Anti-de Sitter, if anything. The extrapolation to de Sitter is still as far as I know an unsubstantiated conjecture, unless there have been serious results demonstrating otherwise.

#### PrashantGokaraju

The Hamiltonian 1/2 (p2 - m2) is conjugate to the proper time τ. If instead we use an arbitrary parameter λ, then we must include a factor e(λ) called the 'einbein', and the action becomes

∫ dλ e/2 (p2 - m2)

where e(λ) dλ = dτ. The inclusion of this factor makes the theory invariant under reparametrizations τ → λ. In general relativity, the motion of a particle in a gravitational field is independent of its mass. So the value of m is arbitrary in the lagrangian. This is analogous to the arbitrary character of e. In a sense, m = 1/e. In spacetime, we can introduce an einbein field j+. When we go from minkowski to general coordinates,

η → j+ j-η

This j is to be considered as the canonical conjugate of a self-dual vector potential A+. The j or the A can be considered to be the gravitational field. In terms of these 'new variables', the Hamiltonain of general relativity is

C- = ϵ j+ j- F

This is called the Hamiltonian constraint. There is another quantity called the diffeomorphism constraint C+ = j- F. The equations C+ = C- = 0 are satisfied by so-called 'physical' states, and belong to Hphys.

#### mitchell porter

@PrashantGokaraju says a number of intriguing intuitive things in this thread. They deserve a broader response, but I will just focus on two things.

First, on de Sitter and Chern-Simons, this comes from Smolin 2002 and drew a response in Witten 2003.

Second, @Demystifier is correct that "space dimensions in the bulk could be equivalent to time dimensions on the boundary" is a wrong guess. In AdS/CFT, there are two kinds of extra bulk dimension that need to be accounted for: the extra "radial" space dimension of AdS per se, which gives the bulk one more dimension than the boundary, and then enough compact space dimensions to reach the 10 dimensions of string theory or the 11 dimensions of M-theory.

It's clear that the radial dimension has something to do with scale, and renormalization group flow, in the boundary theory. The similarity between AdS geometry, and the tensor networks of MERA (multiscale entanglement renormalization ansatz), is now well-known, although I believe work remains to be done in rigorously relating the two.

As for the compact extra dimensions, what they represent, in terms of the boundary theory, is a lot harder to generalize about. Five years ago, David Berenstein, who must be one of the top people to have studied this topic, gave three "sketches" of how the geometric description of objects in the bulk, corresponds to internal degrees of freedom of objects (e.g. "droplets") on the boundary.

I suggest that the way to think about this, is to recall that points in the space-time of the CFT, correspond to asymptotic "points" on the boundary of the AdS theory; and that correlation functions between points in the CFT space-time, equal scattering amplitudes for objects that asymptotically enter and exit AdS at the corresponding "points" on the boundary. That may sound a little abstract, but look up what a Witten diagram is (it's the AdS/CFT counterpart of a Feynman diagram) and it may become a little clearer.

The creation of an inbound or outbound object, at the boundary of AdS, corresponds to a source or sink in the CFT. The objects in the AdS theory should all be strings or branes, and the sources in the CFT are gauge-invariant combinations of CFT operators, localized at a point. A particular operator combination, creates a particular AdS object, moving into the holographic radial dimension. So the compact extra dimensions, must somehow have to do with parameters of these CFT operator combinations.

I am sorry for being technical and vague at the same time, and for going on at such length about AdS/CFT, in a thread which started out as being about time in QFT.

Actually, I would like to comment on a third thing that Prashant said: "The open strings and closed strings are in some sense equivalent because of the equivalence between gauge theory and gravity." Well, there are dualities between open and closed strings, the simplest of which is that a cylinder can be regarded as a closed string evolving over a time interval, or as an open string evolving in periodic time. And there may even be a way to "apply" this to gauge/gravity duality, in the context of black D-brane stacks, where the gauge theory described open strings attached to the D-branes, and the gravity theory describes closed strings within the event horizon of the stack. But I do not remember how this works.

#### PrashantGokaraju

The current j is conjugate to A in the same way x is conjugate to p. This means that in a sense

j = d/dA

The addition of the term jA to the Lagrangian is a Legendre transformation. The Legendre transformation allows us to consider L as a function of p instead of v. This is related to the fact that the kinetic energy

T = 1/2 (dφ)2

is a function of v = dφ instead of p.

H = pv - L

is the hamiltonian. This is the sense in which the Lagrangian can be considered to be a Hamiltonian.

#### Auto-Didact

The current j is conjugate to A in the same way x is conjugate to p. This means that in a sense

j = d/dA

The addition of the term jA to the Lagrangian is a Legendre transformation. The Legendre transformation allows us to consider L as a function of p instead of v. This is related to the fact that the kinetic energy

T = 1/2 (dφ)2

is a function of v = dφ instead of p.

H = pv - L

is the hamiltonian. This is the sense in which the Lagrangian can be considered to be a Hamiltonian.
That the direct application of the Legendre transform to a function gives another function is only applicable contingently and not completely necessarily, i.e. only given that certain mathematical conditions are met w.r.t. both explicit functions.

Often when presenting a function implicitly - as functions are generally presented in the context of physics - most physicists just tend to outright assume without any justification that these conditions are met, purely for reasons of mathematical tractability, when in actuality whether or not such conditions are met is unknown and constitutes an open problem in mathematics.

Therefore, while your argument may be somewhat convincing at the level of rigour for many - if not most - physicists, your premise is actually de facto logically false and worse, mathematically flawed.

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