Kaluza Klein and Yang-Mills Topology

This is a hodge-podge of K-K related ideas (many are questions!) such as yang-mills and topology :

• A little introduction to the gauge principle. The idea is very simple if you are familiar with the classical physics of rotating frames. In that scenario, we choose a coordinate frame which is rotating (for example, the earth). Now, when we measure a velocity there are two pieces : the coordinate value changes with time, and the axes change with time. In particular, if e_i is the i'th coordinate unit vector :

  (a brief notation convention intro, since this is pedagogy:)
  x_i is the ith component of the vector x, x^i is the superscript version
  x^i y_i is contracted to be a scalar (dot product, sum on repeated indices
  d_t is the t-derivative
  coordinate bases are chosen in a local frame
  

  x = x^i e_i
  d_t x = (d_t x^i) e_i + x^i ( d_t e_i )
  d_t e_i = W_ij e^j
  d_t x^i = v^i
  d_t x = v^i e_i + x^i W_ij e^j
  d_t x = v + (x W)
  

  We see the true time derivative of x involves the velocity in the local frame, and a curvature from the W "connection" (seeing the future) which tells us how the coordinate bases move in time. (of course, if you can understand my notation, then this is review for you).

  Now, let's talk about particle physics. We imagine the electron (or what have you) is a little bar which points in some (very small) 5th dimension. The equations of motion of the electron actually apply to the *head* of this little bar, but the tail lives in our spacetime. So, you can imagine the electron whipping about in spacetime; it has an ordinary velocity, as above, but the coordinate basis in this extra dimension may also be rotating (or what have you) : this will create apparent forces as seen in spacetime.

  It's easy to develop this by analogy to the above. An electron (Q) is a wavefunction (q) plus some local coordinates : Q = q^i e_i , Thus, in a d_u (u is a 4-index, space or time) change:

  d_u Q = (d_u q^i) e_i + q^i (d_u e_i)
  d_u e_i = A_ui^j e_j
  

  A is the connection : it as a lower index "u", and two matrix-indeces i and j . A tells us how much the little bar moves in spacetime as it follows a straight path in its fifth dimension. (imagine an electric race car on a little track : it tries to go forward, but the path is curved; the velocity is just straight forward always, but A_ui^j tells us how the path curves : as measure in the frame of the track, the path is straight, as measured in spacetime, A acts like a force). A_ui^j is how much the electron is "rotated" into the j direction, from the i direction, as it moves in the direction u. (here i and j are indices in the extra 5th dimension, u is in spacetime). In the special case where there is only one basis vector (a 1-dimesional extra space) the indeces fall off :

  d_u Q = d_u q + a A_u

  (we will use this notation even when the indices should still be present, because we take A to be matrix-values in extra-space (like the momentum of inertia matrix)). We summarize this with the covariant derivative :

  D_u = d_u + A_u
  d_u Q = D_u q
  

  So this whole processes has just been so that we can commute that capital from Q to D :^) No, seriously, we don't know what Q is, but we do know what q is, which is why we have to do this.

  It is important to note that Q is an invariant : it is a scalar, coordinate independent (in the extra dimensions), but q is not. That is, Q = q^i e_i . Any change in coordinates e_i can be compensated by a change of q^i . This is called choosing a "gauge". (why don't we call it choosing coordinates in the extra space? It's an unfortunate historical accident : the name was chosen before people understood what was going on. It's just the same as picking the coordinates you do your classical physics in). Now, just like good physics should be independent of how you choose your coordinate system (which gives momentum conservation, btw) so, physics must be "gauge invariant" , which means independent of how you choose your gauge (coordinates for how the little bar lives over spacetime).

  So, let's explore gauging. A coordinate transform is :
  e_i -> g_i^j e_j which is compensated by
  q^i -> h^i_j q^j where g and h are chosen such that hg = gh = 1 (in matrix notation) so that Q is invariant. The only subtlety is that A is also changed. Recall:
  d_u e = A_u e
  Is changed to : d_u (g e) = B_u (g e)
  B_u e = h d_u (g e) = h (d_u g) e + hg d_u e
  B_u = h (d_u g) + A_u
  We see that A_u has been shifted by the derivative term h(d_u g). (again, in the special case where there is only one direction, h and g are scalars, not matrices, and this is simply (A_u += d_u f) for any function f). Now, since Q is a scalar, d_u Q has not changed. What about D_u q ? First, let's look at d_u q :

  d_u q -> d_u (h q) = h d_u q + q d_u h

  That's not good. But now for D_u : D_u q -> h d_u q + q d_u h + hq B_u = h (d_u q + A_u q) + hq (g d_u h + h d_u g)

  = h D_u q + hq (g d_u h + h d_u g)

  This last term is actually zero. Since gh = 1, d(gh) = 0 = g d h + h d g , shows it. thus

  D_u q -> h D_u q

  Which means, for example, that [ e^i (D_u q)_i ] is invariant. More importantly :

  ( q* D_u q ) is gauge-invariant.

  where the * means adjoint (transpose makes it a column vector to dot with the row q).

  We won't go into more detail now. This has been a magical ride. The "connection" A is nothing other than the photon 4-vector potential of Maxwell (a student of classical physics could appreciate this as a consequence of the fact that Maxwell's equations are invariant under gauge transforms A += d_u f , as we demanded here). The "gauge principle" shows that only a few types of terms can be used in physics : those that are gauge invariant; we've shown how to construct them. More importantly, it shows that physics is all geometry. Particles always move in straight paths; the gauge potential, the connection A, simply tells us how a straight line for the particle looks in spacetime.

• Why do charges radiate? Ok, that seems silly, but let me get into it; if you understand the answer, you know most of modern physics.

  
  >From a theoretico-inductive viewpoint, I can't
  understand why Maxwell fields (phase gaugons)
  should have sources.  Let me state some things:
  
  dF = 0 is just Bianchi, it means F is a curvature
  
  fermions (matter) "couple" to the "gaugon"
  (I should say, are sections of the fiber in which
  the photon is the connection) because of the
  "minimal coupling" prescription in the LaGrangian:
  
  psi* ( i gamma d ) psi -> psi* ( i gamma D ) psi
  
  this is entirely geometrical, and just expresses the
  fact that A is the parrallel transport of the fermion.
  
  But now we have an odd coincidence.  By changing the
  fermionic term in the lagrangian, we've changed the
  equations of motion for the photon as well!  When
  we vary with A, we now get an "accidental"
  contribution of
  
  	*d*F = psi* i gamma psi
  
  The standard current.
  
  But how can we understand this geometrically?  Just because
  F is the curvature of the fiber that psi "lives in", why
  should it follow that psi is a "source" for F ?
  In particular, the equation
  
  	*d*F = j
  
  says geometrically that when a section is following a
  holonomy, the infinitesimal volume around that point contains
  areas of infinite curvature (the integral version of Gauss's
  law).  Why does a probe necessarily interact with the thing
  it is probing?  (I know several principles demand this, but
  I want to derive the principles)
  
  (BTW we can understand the "accidental" forces induced by
  the LaGrangian prescription as a statement of Newton's
  law on "equal and opposite" : whenever body A acts on body B,
  inducing a term in AB in the lagrangian, variation in the
  other variable induces a force on A from B.)
  

  Let me rephrase this for (semi-quantum) gravity, where the geometry picture is more familiar. Given that we have the Christoffel connection, and matter (sections of the tangent bundle of the spacetime manifold) are parrallel-transported by this connection (this is simply the statement that d is replaced by the covariant derivative in the equations of matter motion) : why is matter a source for gravity? As in the last post, I can see it from an "accident" in the action principle. As Baez points out in "Gauge Fields..." this accident is a happy one : it means that the equations of motions with sources lead to conservation laws (energy-momentum from gravity, and charge from maxwell); but this is really just another way of stating the action-principle- accident (combined with the Noether method).

  As you note, this has a really nice conceptual interpretation. The Lagrangian formalism has, built right into it, the principle that "if X affects Y, then Y affects X". Perhaps the most famous application of this principle was Newton's realization that when the earth pulls the appled down, the apple pulls the earth up. The second most famous application was Einstein's realization that since the metric on spacetime affects the motion of matter, the motion of matter should affect the metric. This follows automatically from the Lagrangian formalism, *if* we treat the metric as a dynamical variable. So perhaps Einstein's real insight was that we *should* treat the metric as a dynamical variable. More generally, Wheeler has phrased the principle of "background-freeness": we should treat ALL tensor fields as dynamical variables --- we should not have "background structures" around which affect things but remain unaffected. The third most famous application is the one you note: if gauge fields affect the motion of charged particles, the charged particles must serve as a source for the gauge fields, at least if the gauge fields are treated as dynamical variables. It's also amusing to note that the principle "if X affect Y, then Y affects X" was condemned as a heresy by the archbishop of Paris in the middle ages. The reason is obvious: it limits God's omnipotence and makes him basically just another part of the universe. >But how can we understand this geometrically? I think that's the really interesting question here: how can we understand more deeply the *geometry involved* in this particular example. It's cool, for example, how Bianchi nails down dF, while the field equations nail down *d*F, leaving only the harmonic part of F unspecified --- which of course describes "pure radiation".

• Anomalies and whatnot are full of pretty pictures and relationships. Let me quote two relationships from Yang-Mills that I don't fully understand :

  • Chiral Anomaly (d^u j5_u) = Instanton Density = F/\F

    (I've normalized so all the constants drop out). Now, there's a whole rot about zero modes and atiyah-singer theorems and whatnot, but it still remains that the direct relationship of chiral anomalies and instantons is unclear (to me). T'Hooft and others have claimed that "instanton interactions cause the anomaly", and that is fine (in fact, the instanton ("theta") term in the Lagrangian can be thought of as an effective chirality-changing term (that is, a psi-bar-psi term)) - however, they (he) use the above relation to justify that claim, not the inverse (explaining the above from some physics).

  • Dilaton Anomaly (Trace of T_uv) } = F/\*F

    Minimum { Dilaton Anomaly } = F/\F

    These two statements are simply related by the fact that Min{F/\*F} = F/\F (which was how we got the instantons in the first place). Perhaps it would be nicer to simply say that : when F is self-dual, the Dilaton and Chiral anomalies are equal. This, I do not understand.

  • It is tempting to say that these equations are simply coincidences, because we simply do not have too many things to go around (for example, in binary arithmetic on single bits, I can do lots of complicated things that all give the same answer - simply because there are only two possible answers : 0 and 1) To some extent we have the same situation here : in 4-d, F/\F and F/\*F are all we have to play with. Someone let me know if we can do better.

• The wormhole picture of charge is quite pretty, and gets prettier the more you think about it.

  First a brief review. We replace the yang-mills theory with sources with a theory of only free-space fields (DF=0 and *D*F=0). Instead of charges we have topological holes in our "empty" space : handles : tubes/wormholes from one point of space to another. Then field can travel through these handles, which will make one end of the tube a source for field, and the other a sink; this is what we typically call a particle and an anti-particle. From afar these hole-mouths appear identical to particles.

  Now some interesting stuff. First, we assert that it is impossible to observe charges - only their emissions are real; hence a charge is measured only by its Gauss-Law flux as a function of the location and size of the enclosing sphere. In this sense and ordinary charge has uniform magnitude of emissions no matter how close we get (you can never quite reach the delta function). However, with the wormhole charge (WHC) this is not true. As you get closer, spacetime becomes curved to go through the hole, so the apparent distance becomes nonlinear in the actual distance - ie. the apparent distance to the charge never gets smaller than the throat-radius of the hole regardless of how close you move.

  Particle annihilation is also amusing. It consists of the throats of the wormhole moving ever closer, so that the volume of hyperspace in the handle shrinks. Eventually the handle pinches off, and the bridge of spacetime between the throats is squeezed down to zero. This is a topology-changing process, something which we usually think of as forbidden. Similarly, pair creation consists of a bubble in hyperspace stretching out, touching the spacetime manifold, and opening up holes in it to the inside of the bubble. Vacuum screening is an interesting thing in this context - any wormhole mouth is surrounded by other tiny wormholes which spontaneously form from bubbles of this type; some of the field spewing from each wormhole mouth travels not through spacetime but into the cloud of small wormhole mouths nearby.

  The right way to study wormhole charge seems to be a Kaluza-Klein theory. In this scenario we have "energy" (fiber curvature) all over spacetime, and handles with severe spacetime curvature; the general relativity equations provide the dynamics of the wormholes. We can require that Einstein's equations preserve the wormholes, but allow their motion; this might constrain the charge & shape of the fields (from "particles"). We would also then get the dynamics of the wormholes. Pair creation and anihilation could be dealt with classically by assuming that spacetime was covered with infinitely many infinitesimal handles, which could expand/contrat as dictated by Einstein.

  The "fermionic" (exclusion-istic) nature of conventional particles seems to be a problem in this context. There is no principle to govern the statistics of two wormhole mouths - we can imagine them colliding and even joining completely to make a double-charged mouth. They have no statistics because they are not identical - each is connected to a different "anti-particle" (other end of the wormhole tube) which in principle makes them distinguishable.

• I'm curious if anyone has ever investigated the character of the physical yang-mills theory constructed from the connection on a non-"group" manifold. (perhaps even not a smooth manifold!) In most cases it would seem such a theory would either be trivial or too complicated to do calculations with, but perhaps there have been some general arguments made about the character of such a theory? Is there a no-go for non-group yang-mills? (of course supersymmetry provides some sort of an example of a non-Lie yang-mills, but it seems there are even more general types of fibers which we could put over space). (also any non-smooth manifold would surely give rise to some kind of infinities, but these are nothing new to the quantum theorist)..

  In a similar vein, I note yang-mills is the gauge theory of connections of fibers over spacetime, while general relativity is the gauge theory of the spacetime connection itself : now we generally assume that the tensor product of these two theories covers all cases, but it seems to me there is a non-trivial additional case. That is, there may be a connection in a manifold which shares the tangent bundle of spacetime partially, and some fiber bundle over spacetime; this would trivially reduce to some sort of YM + GR , except in the case where there is some non-trivial topology between the fiber-part and the spacetime-part of the connection; that is sections of some manifold M which coincides partially with spacetime, and has some complicated structure; perhaps being mobius, or having some sort of knotting between the spacetime and fiber part... (?)

• A question, if you please :

  we can write Einstein's Eq. , with the stress-energy tensor from Yang-Mills ; this seems extremely general, since the standard model is nothing but yang mills (we'll ignore massive particles that count as the source for these). So,

  [ R_uv - 0.5 g_uv R ] = 8piK [ F_uw F^w_v + 0.25 g_uv F^ab F_ab ]

  Now this seems too beautiful not to have some powerful direct interpretation. On the left we have the curvature of space-time (contracted in various ways), and on the right we have the curvature of the fiber-bundle for the yang-mills field. So, in some space, (E in your notation) we just have a space M which is curved by R, and a space (E/M) which is curved by F. In other words, if we had a "vector" (in M) of "sections" (on E) (apologies if my nomenclature is sloppy, what I mean is a tensor product of the basis of vectors d_i and the basis of sections, e_i, such as [X^ij d_i e_j]), then when we parrallel transport this vector of sections, around a small loop (for example), the vector part will be changed by R, and the section part will be changed by F - and these changes are not independent !!!!

  Anyway, now to the question : is there some way to directly interpret this fact geometrically? It seems that some inherent topological relation between the spaces M and (E/M) is implied. Are there some sources in the literature which you could recommend?

  Kaluza-Klein is one way

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