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A String vacua and particles?

  1. Dec 4, 2017 #26
    I'm going to ask a very naive question (so apologies in advance): I'm wondering why, in Feynman diagrams, the internal lines represent integrals over all possible momenta, and likewise, in string theory, why we sum over all possible word sheets.

    In a scattering experiment, the incoming and outgoing particles are taken as having definite energies, yes? Now I may not be able to measure those energies (HUP), but are they nonetheless fixed? If not, then I can see the possibility of having to integrate over all possible momenta, but I don't think I'm correct...
    Last edited: Dec 4, 2017
  2. Dec 4, 2017 #27
    There are a lot of questions accumulating in this thread. I will start with a few big-picture remarks.

    Field theory and string theory have a mathematical aspect and a physical aspect. The mathematical aspect usually includes some procedure of calculation. The physical aspect involves physics concepts like matter, force, space, time... Discussing the physical aspect of quantum theories brings well-known problems because of concepts specific to quantum mechanics, like superposition, complementarity, and the emphasis on "observables" rather than "physical facts".

    In #14, @asimov42 asked about the ontological status of virtual objects in string theory. This is hard to address without knowing what approach to quantum ontology in general is being assumed. For example, are we to assume that observables are the only real things? Or are we being asked to say something about reality between "observations"? Or is it really a question about what string theorists think, what their attitude to virtual objects is?

    Strict adherence to the positivism of the original Copenhagen interpretation offers one kind of clarity. There are quantum states, there are observables, and all there is to say ontologically about a quantum state, is what it implies for observables.

    An alternative kind of clarity is offered by a reconstruction of quantum mechanics into a theory in which reality is described by a collection of objective physical facts, in which observation, measurement, etc., play no fundamental role. Bohm offered such a theory, Everett tried to, so do various other schools of thought.

    Perhaps the majority of discussion about quantum mechanics takes place between these poles of clarity. But if we really are going to discuss the ontological status of virtual quantum objects, it would help if the participants explicitly indicated whether they speak as Copenhagen subjectivists (example: Lubos Motl) or as quantum crypto-realists (example: Brian Greene), or otherwise said something about their place on the spectrum.

    Now to some specifics. It was said in #6 that "there is no time-energy uncertainty principle". Well, it's not as straightforward as the uncertainty relations arising from the usual complementary observables, but time-energy uncertainty relations can be derived, e.g. by considering time-of-flight of a particle in an energy eigenstate that is tunneling through a barrier.

    I tried to find the origin of the idea that time-energy uncertainty is to be expressed (or even explained) in terms of "borrowing energy" for a limited time. Via Peter Holland's Bohmian text, "The Quantum Theory of Motion", section 5.3, I found a 1974 paper by Hirschfelder et al which does actually defend this interpretation (end of part III). Holland criticizes it; I have not tried to analyze the original argument or Holland's criticism.

    Neumaier and others have emphasized that "quantum fluctuations" do not refer to something changing in time, but rather to an observable with a range of possible values. These statements seem to come from the Copenhagen end of the "axis of interpretation" that I described. The closest thing to an ontological argument that I see, is the remark that virtual particles only exist in perturbative methods of calculation; obviously they can't be objectively existing objects, because objective existence can't depend on a method of calculation.

    For someone at the other pole of interpretation, someone who is seeking a characterization of objective reality, that argument might be salient but not decisive. So long as no definite ontological picture is presented, a person can still think, nonetheless maybe that is how it is. For example, what if we could keep our virtual objects "on shell"? Would that make them candidates for objective existence after all? Or, what if energy is borrowed from, and then returned to, a realm of subquantum thermal fluctuations? Could we implement energy-time complementarity, in a form where the energy really is borrowed from somewhere?

    So, as difficult as it is to have that sort of discussion and the more mathematical discussion at the same time, I think there can be no real clarity until people at least indicate where on the axis of interpretation they're talking from.
  3. Dec 4, 2017 #28


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    Because of superposition of wave propagation. Look eg in Zee's qft book for an intuitive derivation how superposition leads to path integrals. Or, classically, the Huygen principle for wave propagation.
  4. Dec 4, 2017 #29
    @haushofer I think (maybe) I have a partial grasp of wave propagation ... I was just looking at Zee QFT book. But the 'one loop' corrections in QFT are taken over all momenta (hence the off shell and virtual particles - as @A. Neumaier would say, bookkeeping devices) - the incoming particles are most definitely on shell. So we are integrating (at intermediate points in the perturbation series) over momenta that the incoming particles cannot posses.

    Also, any comments on the energy of particles in a scattering experiment? I understand each particle may be in a superposition of energy states - but that does not mean it may have any energy (e.g., infinite).
  5. Dec 12, 2017 #30
    Let's tackle some of these questions...
    I'll sum up this question as, why go off-shell? The most general answer I can give is: so we can have quantum mechanics, rather than just classical mechanics.

    On shell means obeying the classical equations of motion, off shell means not obeying the classical equations of motion. Being on or off the mass shell is just a special case of this, the case of a free particle with a specified mass.

    A path integral is by construction a sum over a set of possible histories. The histories which strictly follow the classical equations of motion are a subset of measure zero in that set.

    Giving each history a weight of exp(i.action) means that the probabilities peak around classical behavior. But most of what you're summing over is just randomness, and allowing that is an essential part of the framework.
    The usual Feynman diagrams correspond to particular limits of the string interaction diagrams - see what Urs said in #15 about decomposing the string worldsheet history into cylinders and strips.

    Unfortunately I haven't found a paper or textbook which illustrates what I want to say; but scroll down this essay by a string theorist, to the picture which shows four Feynman diagrams and four string diagrams. You can easily see the similarities, but there's something more.

    Feynman diagrams a, b, c correspond to different terms in the perturbation expansion of some quantum field theory, but string diagrams a, b, c are actually different instances of the same term in the perturbation expansion of string theory.

    That's because the diagrams in perturbative string theory are classified by topology, and string diagrams a, b, c all have the same topology - two closed strings in, two closed strings out, and no "holes". That defines a single term in the topological expansion of string theory, and to evaluate it, we have to sum over all worldsheet histories with that topology.

    Some will look like a tall X (a), some like an H (b). Diagram c might look different because the strings seem to change places, in a way that they don't in diagrams a and b, but that's just one way of seeing it. In fact, you can smoothly deform a into c, by taking the vertical segment where the two strings have merged, and twisting it upside down.

    So the point is, there's only one tree-level scattering diagram in that string theory - only one topology to consider, for tree-level scattering - but different parts of the path integral correspond to different Feynman diagrams in the field-theory limit.

    This unification of Feynman diagrams that ought to be distinct, into a single amplitude, was discovered even before string theory was recognized as a theory of strings - see the story of the "Veneziano amplitude".
  6. Dec 18, 2017 #31

    Urs Schreiber

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    Yes, the perturbation series of every non-toy QFT diverges (Dyson 52).

    The modern perspective is that these series are to be regarded as "asymptotic series".

    Concepts like "virtual loops" only exist in perturbation theory. The renormalized Feynman perturbation series is finite at each loop order , and so is the string perturbation series (not proven rigorously though, i suppose) but both still diverge when summing up all loop orders.

    See also the string theory FAQ at Isn't it fatal that the string perturbation theory does not converge?
    Last edited: Dec 18, 2017
  7. Dec 18, 2017 #32

    Urs Schreiber

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    Just to amplify that this is not special to string theory and that the same statement applies also to QFT: The Feynman amplitudes in QFT may be understood as coming from the 1d worldline field theory of a quantum particle in direct analogy to how the string scattering amplitudes come form a 2d worldsheet field theory of a quantum string.

    This fact (or insight) is called worldline formalism of QFT, due to Bern-Kosower 92, Strassler 92. It makes manifest how perturbative string theory is a straightforward/natural variant of perturbative QFT.

  8. Dec 18, 2017 #33

    Urs Schreiber

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    On the contrary, it's exactly as in standard Feynman diagrams, just with 1-dimensional graphs replaced by 2-dimensional surfaces. That's the very definition of perturbative string theory: Replace the formula for the S-matrix, originally given by a sum over Feynman graphs, by a corresponding sum over 2-dimensional surfaces.

    What is observable about this (in both cases) is the end result of the sum, which is (an asymptotic series of) the probability amplitude for given states to come in from the asymptotic past and for other given states to emerge in the asymptotic future.

    A priori nothing tells you that each single term in the sum has a corresponding physical interpretation. What you keep asking is what the physical interpretation is for each single term in this series. Generally the answer is: It has none.

    But of course if you look at these terms, it appears extremely suggestive, intuitively, to assign physical meaning to them, in terms of "virtual processes". But since this is not what the maths tells you, but just what your intuition tells you, the rule to proceed is the following:

    As long as you find it helpful to think of single summands in the perturbation series as "virtual processes of particle/string interactions" run with it, but as soon as you find yourself bogged down in trying to make concrete sense of this intution, let go of it. Because, it's just that: an intuitive picture that only carries so far.
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