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Spacetime in quantum theory

  1. Apr 14, 2004 #1
    As I understand, QFT is formulated on a Minkowski background spacetime. This spacetime is not dynamical.

    So, what a priori reason is there for formulating QFT on a flat background spacetime? Is it a question of empirical evidence? Or of mathematical utility? Or both, something else? I have heard/read that a transformation from one classical background spacetime to another does not necessarily correspond to a unitary transformation, and so we have inequivalent quantum theories. Is this true? Is there a restriction on what background spacetimes we are able to formulate our QFT on?

    What is the predominant view of the ontology of this background spacetime in QFT? If there is a fixed inertial reference frame, then we are granting ontological significance to the spacetime and its individual points. This conflicts with general covariance. If the background spacetime was not fixed, or at least we could choose any reference frame from an equivalence class, in what sense could the background spacetime be regarded as not real, as a fictitious mathematical entity?


    If the background spacetime is regarded as real, is it regarded as a container space, in that it can be colocated with the dynamic objects of QFT? Ie: An object can be located at a spacetime point, but different spacetime points can not be located at the same spacetime point.
    Last edited: Apr 14, 2004
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  3. Apr 14, 2004 #2


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    'As I understand, QFT is formulated on a Minkowski background spacetime. This spacetime is not dynamical'

    Yes, in other words its a special relativitistic theory, not a General relativistic one.

    'So, what a priori reason is there for formulating QFT on a flat background spacetime?'

    Its a lot easier to deal with for one. For two, geometry change is expected to give contributions many orders of magnitude weaker than the other field forces, on appropriate length scales (say what you would measure in a lab).

    'I have heard/read that a transformation from one classical background spacetime to another does not necessarily correspond to a unitary transformation, and so we have inequivalent quantum theories'

    Yes, thats a big problem naively for quantum gravity, but there seems to be ways around this.

    'If there is a fixed inertial reference frame'

    No, that would already violate special relativity, and our nice experimental results. Some physicists play around with this idea, but I think its wrong and so do the majority of working scientists..
  4. Apr 14, 2004 #3
    Thank you for your reply Haelfix, that answers a few questions. A few more questions in response to your comments.

    What ways in particular are you referring to?

    So there is an equivalence class of background spacetimes that QFT is formulated on? I would argue that, in that case, given that we can label two points equally well in a different manner this destroys their ontological significance (basically the same argument as general covariance) and hence we can regard the background spacetime as a fictitious mathematical entity. Do you agree?
  5. Apr 14, 2004 #4


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    QFT requires the Minkowski signature, you are free of course to rescale these things such that the symmetries of the system are conserved. General covariance, allows you to pick inertial systems where the curvature identically vanishes. So you can write a QFT in such a place, but you will lose all global properties of interest.

    If you really want the full global theory, naive QFT cannot deal with many backgrounds of interest to general relativists, such as ones with large spatial curvature (Schwarschild metrics for instance).

    To get a qft that could potentially work there, you need a theory of gravitons (eg a quantum theory of gravity). See String Theory for instance, in some limit it reduces to just such a qft.

    You can naively try to extend field theory into curved space, just write the path integral for a graviton, and its vacuum diagrams. There will be curvature tensor parts, and all that nice familiar stuff from GR. You might worry about nonrenormalizability, and other nasty things like conformal anomalies. But if you cut off your series, you will get an answer.

    In that setting (I dont know much about String theory), the unitarity problem arises in the sense that it is difficult to define the vacuum of the system identically, and its time evolution has to be treated very carefully to avoid disasterous ambiguities. But there is a song and dance, about how to fix this (warning its highly technical).

    AFAIK, the problem with String theory and unitarity, is more in the topological sense. Topology change, should apriori kill any meaningfull sense of unitarity.

    I don't know what that means there, or if its really a big deal. Models we use to describe the real world, can arguably always be considered fictitious. Depending on who you ask, you will get a different vision of what is truly fundamental.

    I tend to think algebra is, but if you ask a String theorist, he'll poo poo that.
  6. Apr 19, 2004 #5


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    Maybe Haelfix will confirm or elaborate on this: I believe one reason Minkowski space is so natural, from a GR standpoint is that it is the simplest solution to the Einstein equation for the gravitational field.

    To restate a something well-known to you, just to have it explicit here: solutions to the field equations--gravitational fields---are metrics.
    The Minkowski metric is the solution you get from the GR equations when there is no matter or energy in the universe to cause curvature, no initial curvature assumed, no expansion etc.

    So Minkowski space is somehow like "GR zero" or the most basic possible gravitational field, the geometry you get when there is nothing in the universe.

    since in the real universe matter is distributed pretty sparsely on average, what could be more natural to take as an approximation?

    so as you say there is mathematical utility in its favor and empirical rightness (on the whole), and there is also this naturalness----it is the most vanilla of
    all gravitational fields.

    I trust Haelfix will tweak this if it is askew. :smile:
  7. Apr 20, 2004 #6
    The reason we work with flat spacetime qft most of the time is because we are trying to figure out the rules of the fundamental particles without adding the complication of curved spacetimes (it's hard enough without them!)...and practically this works fine since in our particle accelerators since spacetime curvature is small for that region.

    But QFT can be constructed in curved spacetimes, as naturally as in flat ones. A big difference between the two is that the vaccum states in Minkowski space QFT are Poincare invariant (the Poincare transforations are Lorentz transformations and space-time translations...also, Minkowski space refers to flat spacetime), whereas in qft in curved spacetimes, the vacuum is only Lorentz invariant in a local reference frame, and certainly not globally Poincare invariant. This means that observers traveling at different speed with respect to each to each other will not see the same vacuum: different observers see different particle content in the vacuum relative to each other...this is called the Unruh effect. By the same token, this also means that in the presence of large enough spacetime curvature (like a black hole), if you are hovering above it you will see a non-trivial particle content of the vacuum...this is called Hawking radiation. Both are consequences of the fact that in GR, you have invariance under local coordinate transformations, but not of global Poincare transformations.

    You don't need quantum theory of gravity (meaning, no need for gravitons or anything more subtle) to formulate qft on curved spacetime backgrounds. See Wald, "QFT on Curved Spacetimes...." for example.
  8. Apr 20, 2004 #7


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    Although I only do know a little bit about QFT, I took a look to some references about QFT in curved spacetime and it was a surprise for me that the first problem one has to face is how to define a complete set of functions as a basis of solutions for the Klein-Gordon equation. Different observers seam not to agree on which negative modes or positive modes to choose. I assumed this leads to the weird conclusion that the different chooses of spacetime coordinates imply different views of what matter and antimatter is. Is this correct?

  9. Apr 21, 2004 #8
    There is a technical way of seeing anti-particles as being equivalent to particles traveling backward in time, so with this intuition, a Lorentz boost should not take anti-particles to particles and vice versa, even in a curved spacetime (such a transformation would have to map points in the forward light cone to the backward light cone and vice versa).
    The problem is close to what you described: it lies in defining a unique space of *positive frequency* solutions (and consequently, but seperately, a space of "negative frequency" solutions). But defining the space of these solutions to, say, the Klein-Gordon equation is supposed to lead to the Hilbert space of states, and so ultimately we find a freedom in choosing a Fock space. The Fock space is the space of states in the "particle number representation" that tells you how many particles of particular quantum numbers you have. Due to there being no preferred Fock space, there is no absolute notion of the particle content in a certain region of space.
    This should not be too troubling if you keep in mind how much we have done away with the notion of particles being little bb's bouncing around since the early 1900s. Special relativity combined with quantum mechanics taught us that quantum fields, not particles, are fundamental. This is in the spirit of nature only being relational (in the sense that only the relationships between different observers in space and time make sense, whereas absolute notions of things take a backseat as approximations).
  10. Apr 26, 2004 #9


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    Hi Javier,

    Yes, I agree with you... At least for classical globally hyperbolic spacetimes. I wouldn't call it 'naive' qft though, precisely b/c things are no longer under the poincare group. Lets face it, the Hilbert space of the canonical commutation relations is no longer uniquely defined, and the only way to make sense of the S matrix is to use the algebraic formalism (and then only when you have good asymptotic states).

    Also, without gravitons the theory is sick, and you end up with horrendous divergences that have no cure, even with smeared out operators.

    I was a little troubled mathematically by some of Wald's ideas, (especially with regards to Hawking radiation) but thats the price you pay at this level of the game.
  11. Apr 26, 2004 #10
    How about spacetime and the Vonnegut theory? :P Slaughterhouse Five is wacked up.. Does anyone know where he got those ideas from for the Traffabadors and etc? Anyways...
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