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How useful is topology in theoretical physics?

  1. Nov 24, 2007 #1
    How useful is topology in theoretical physics?

    By topology, I mean the contents of Munkres book, Hausdorff spaces, homeomorphisms, etc. It seems to me like topology is totally a mathematical construct since the idea of an "open set" in an abstract space seems to have no "physical" meaning outside of Euclidean spaces. So do any of you theoretical physics actually use what is in Munkres?

    BTW: I am trying to decide whether to take the semester of a topology course and I am really not wanting to for the reasons above.
     
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  3. Nov 24, 2007 #2

    quasar987

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    The content of (the first half of) Munkres is the basics of the basics in topology.

    And a differentiable manifold is just a topological space with a differentiable set of charts on it.

    Now differentiable manifolds are all over physics. In fact, they're all over everything.

    But to name a few instances in physics, classical mechanics has a formulation in terms of a simplectic manifold. According to Einstein, spacetime is a riemannina manifold. In unification theory, people are giving some kind of physical meaning to connections on vector bundles.

    So you'll definitely want to get a good grasp of what a homeomorphism is, and what topology is really about besides 3 cold axioms.
     
  4. Nov 25, 2007 #3
    Topological quantum computing is one of the most promising schemes for fault-resistant quantum computing. I can't comment on the understanding of topology needed for it as I have not studied it, nor topology for that matter (although I plan to take a class on it next semester).
     
  5. Nov 25, 2007 #4
    I have been taking GR and QM and I definitely see why differential geometry and manifold theory is important. But topology seems like something else. I mean give me a physical example of a "one-point compactification" or when you would use the "Tychanoff Theorem" or where I can find a locally compact Hausdorff space that is not metric? It seems like mathematicians came up with some of this stuff just out of sheer boredom.

    Anyway, we just did the first half of Munkres this semester and next semester we do the second half. Are you saying the second half of Munkres is more useful i.e. applicable in physics?

    So, I guess my question really is: I want to get a PhD in theoretical physics and do research in string theory, quantum field theory, quantum computing, and the like. If I don't learn the second half of Munkres now, do you think I will regret it and need to learn it later in my career?

    I want to take two more courses and basically the only other math or physics course that fits into my schedule is topology. So I was going to just not take topology and take 2 computer science courses instead: an Operating Systems class and an Algorithms class.
     
    Last edited: Nov 25, 2007
  6. Nov 25, 2007 #5

    quasar987

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    The second half of Munkres is Algebraic Topology. I'm almost certain string theory uses that somewhere, it's such a useful tool in the study of topological spaces.
     
  7. Nov 26, 2007 #6
    don't know about you but the topology option sounds more interesting than the operating system class, ofocurse it will be harder.
     
  8. Nov 26, 2007 #7

    Chris Hillman

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    Topology is your friend!

    Topology is an essential core topic of mathematics, and physical discourse cannot proceed without constant appeals mathematical reasoning.

    Topology has many inspirations, but it might be useful to compare and contrast just two:
    • much of topology (particularly geometric topology and much algebraic topology) concerns the topology of manifolds, including finite dimensional Lie groups.
    • much of the rest of topology (including many topics in the book by Munkres, including the Tychonoff product theorem) is motivated by considering function spaces , probability spaces, etc., as topological spaces (typically infinite dimensional).
    I hasten to add that these divisions are not hard and fast; topology is a unified subject and the interconnections between topology of finite dimensional manifolds and topology of large function spaces are many and intricate.

    (Mathematical adventurers will be intrigued to learn that much topology nowadays is motivated by logic! And I for one foresee applications to dynamical systems in this work.)

    If you really master this stuff you'll always be grateful for the opportunity to take a solid course in "general topology". A solid grounding will make it much easier to appreciate topological nuances in analysis and the modern theory of differential equations (functional analysis, operator theory, measure and probability theory are all needed for quantum mechanics, ergodic theory, and other core topics).

    The one point compactification of the plane is ... (fill in the blank). This came up just the other day in my remarks on stereographic projection.

    [EDIT: And it came up just the other day elsewhere; see this query.]

    Tychonoff.

    One of the most useful theorems in mathematics! "Nice" metric spaces tend to be compact; more generally, "nice" Hausdorff spaces tend to be compact. But large function spaces typically are not compact. (As one punning slogan has it, "Nothing finer than a CH space!" :wink:) If you look through a bunch of analysis and topology textbooks and write down the proofs of all the theorems whose proofs mention it (good exercise, BTW!), you'll see that one reason the Tychonoff theorem is so usefull is it guarantees compactness.

    Some pointers:
    • Haar measure is needed for all kinds of applications of Lie groups in physics,
    • Riesz Representation Theorem, one of the core results in functional analysis.
    • Gelfand Theorem, a core result in the theory of C-* algebras (the foundation of geometric quantization; see this introduction by master expositor John Baez) and noncommutive topology; the basic idea here is to generalize the duality between statements about LCH spaces X and statements about the semisimple commutative Banach algebra of continuous asymptotically vanishing functions on X (this program has far-reaching implications for physics).

    If you look through a bunch of analysis and topology textbooks and write down the statements of all the theorems which mention "locally compact Hausdorff (LCH) spaces" (good exercise, BTW!), you'll see that one reason why (fill in the blank for the appropriate notion of morphism) among LCH spaces form such an important category is that it enjoys good "closure" properties, and offers many of the benefits which accrue from compactness.

    For an example of an LCH space which is not a compact metric space, you can search the Questions in Topology from the Topology Atlas, where you can Ask a Topologist :smile: See also Lynn Arthur Steen and J. Arthur Seebach, Jr, Counterexamples in Topology, Springer, 1978, for zillions of counterexamples.

    Trust me, stuff you see in mainstream textbooks in core topics like topology almost never has the character of a "fantasy invented out of boredom", it is there (chosen from many competing topics in this huge, huge, huge field) because it has proven extremely useful in a great variety of disciplines. For example, harmonic analysis unifies large swathes of Fourier analysis, representation theory, and invariant theory, and is essential in many parts of theoretical physics. (Just look for physics eprints mentioning vector and tensor harmonics, for example! And that's just the trivial stuff!)

    [EDIT: From Mathworld:
    Willard (1970: a compact Hausdorff space X is metrizable iff [itex]\left{ x, x \in X \right} = f^{-1}(0)[/itex] for some continuous function.]
     
    Last edited: Nov 26, 2007
  9. Nov 26, 2007 #8
    I've not studied string theory at all... but from what I've read, topology is extremely important because you work in 10+1 dimensional space. See herefor example. It's also important in the study of black holes.
     
  10. Nov 26, 2007 #9

    Chris Hillman

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    Not because of the dimension of the space, but because string theory uses a great of machinery from geometry and analysis which in turn rest upon topology. The topology needed for studying particular spacetimes studied in classical gtr is comparatively trivial, but "serious topology" is again neeed for the "serious analysis" needed for function spaces which arise in the subject, e.g. "the space of solutions to the EFE" is infinite dimensional and arguably the subject of gtr, period, full stop.
     
  11. Nov 26, 2007 #10
    Peek ahead to section 22.6 in your copy of Zwiebach's "A First Course in ST". There you will find the one-point compactification of the complex plane into the -------.
     
    Last edited: Nov 26, 2007
  12. Nov 26, 2007 #11

    Chris Hillman

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    Ooh, you gave it away! But yes, this is of course the example I mentioned. Although under the influence of algebraic geometry I'd say complex line for C, which is closely related to the real plane R^2. So as jimmy says, the complex projective line [itex]CP^1[/itex] is... (fill in the blank). While the real projective plane [itex]RP^2[/itex] is not ... (fill in the blank).
     
    Last edited: Nov 26, 2007
  13. Nov 26, 2007 #12

    mathwonk

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    in some sense, newtons and einsteins idea was to reduce physics to geometry.

    topology is the most basic topic in geometry, underlying all other aspects of it.


    one specific area of topology is representation theory, which used to be very actively studied in theoretical physics.
     
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