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I have often come across this implicit translation of the classical field of a particle of a given spin into a specific tensor field. But I could not locate any literature from which I could learn this.

In a paper of Avrimidi I found this statement,

"The tensor fields describe the particles with integer spin while spin-tensor fields describe particles with half-odd spins"

I would like to know what is the precise mathematical map that he is referring to.

Like if I have a particle of spin s on a space-time manifold M then the classical field of this particle is a section of which bundle on the space-time?

In the same vein I would like to know some expository mathematical reference on what is a "spin connection". I find the usual physics book definition very unconvincing where spin-connection coefficients are just 'defined" analogous to the Christoffel symbols (but with the signs flipped) when the connection is made to act on vierbeins.

Anirbit
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    Incidentally, for physicists, essentially all bundles are trivial and trivialized. Then they write coordinate-full formulas for the transition maps, especially when doing gauge theory. As mathematicians, we know that there are lots of bundles that are not trivializable, although every bundle is locally so. To a physicist, when a bundle is not trivializable, they call it an "anomaly". So anyway, to describe a bundle to a physicist, you only need to describe the fiber. To describe a particle, you need to give a bundle and an action, and this data should be gauge and Poincare invariant. – Theo Johnson-Freyd Feb 26 '10 at 17:18

1 Answers1

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In a nutshell, particles "are" unitary irreducible representations of the Poincaré group, which is the isometry group of Minkowski spacetime on which it acts transitively. Such representations can be constructed using the method of induced representations (cf. Wigner, Bargmann, Mackey,...) as classical fields on Minkowski spacetime subject to certain field equations: wave, Klein-Gordon, Weyl, Dirac, Maxwell, Rarira-Schwinger,... Mathematically these are sections of homogeneous vector bundles associated to certain finite-dimensional representations of the "little group", which is (the maximal compact subgroup of) the stabilizer (in the spin cover of the Lorentz group) of a point on the "mass shell" (=the momenta $p$ with $p^2 = - m^2$, where $m$ is the mass of the particle). The little group for massive representations is isomorphic to $SU(2)$, whereas that of massless particles to a nontrivial double cover of $SO(2)$. Hence massless particles are defined by their helicity (the label of the $SO(2)$ representation from which one induces) and massive particles by their mass and their spin (the label of the irrep of $SU(2)$ from which one induces). The covariant field equations are (the Fourier transform of) the projectors onto irreducible components.

Let's discuss the massive case, since your question mentions spin explicitly.

There are two kinds of irreps of $SU(2)$, those where $-1$ acts trivially and those where it does not. The former case are the integer spin representations whilst the latter are the half-integer representations.

The integer spin representations of $SU(2)$ are contained in the tensorial representations of the Lorentz group, whence this is why the fields for integer-spin massive particles are tensorial.

The irreps of $SU(2)$ with half-integer spin, those where $-1$ does not act trivially, are not contained in tensorial representations of the Lorentz group, since on these representations $SU(2)$ acts by conjugation and $-1$ acts trivially. In order to describe such reps in terms of fields you need to consider spinor fields, which are sections of spinor bundles (possibly twisted by tensor bundles for higher spin fields).

You can read about spinor bundles on any book in Spin Geometry. For example, there's a book by Lawson and Michelsohn of that name. Green-Schwarz-Witten's string theory book (second volume) has a physicsy discussion of this. They will also define the spin connection, which is a connection induced by the Levi-Civita connection on any "spin bundle", which is a principal fibre bundle lifting the bundle of oriented orthonormal frames in such a way that the bundle map between them restricts to the spin covering $\mathrm{Spin} \to \mathrm{SO}$ on the fibres.

It is not hard to show that the connection one-form for the spin connection, when pulled back to the manifold by the lift of a local frame (hence a local 1-form with values in the orthogonal Lie algebra) agrees with the similar expression for the Levi-Civita connection, which is why many books perhaps do not go through the trouble of defining it properly. It also requires introducing quite a bit of formalism, to which many physicists are allergic; although increasingly less so.

José Figueroa-O'Farrill
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  • The interpretation of spin that you gave in the beginning in terms of representations of the little group of massive particles which happens to be SU(2) and it determining a representation of the Poincare group by the idea of "induced representation" is familiar to me from the initial chapters of Weinberg's QFT book. But that the fields have to always be sections of a homogeneous vector bundle determined by this little group representation s new to me. Could you tell me why this should so? – Anirbit Feb 26 '10 at 18:31
  • Could you also elaborate on the statement, "The covariant field equations are (the Fourier transform of) the projectors onto irreducible components."

    Is the "bundle of oriented orthonormal frames" the same thing as saying that on an n-dim space-time manifold I have a principal bundle whose structure group is SO(n) ?

    Then you want to lift this to another principal bundle whose structure group is Spin(n) ? What is the motivation?

     – Anirbit Feb 26 '10 at 18:31
  • Could you give a reference for the tensor representations of the Lorentz group? – Anirbit Feb 26 '10 at 18:39
  • I think this reference may help http://arxiv.org/abs/hep-th/0611263 In fact, authors deal with massless and massive fields in any space-time dimension. The space is the Minkowski space. There are some pecularities of the arbitrary dimensional case, which are not seen in 4d. For example, the spin is classified according to irreducible representations of $so(d-1)$ for massive fields and $so(d-2)$ for massless ones. So the spin in d>4 is no longer a single (half)integer number but the sequence of (half)integers. – Eugene Starling Feb 27 '10 at 08:19
  • I also must say that the Minkowski space is not the only space on which the fields with the spin are considered. Motivatated by AdS/CFT and the superstring theory on $AdS_5\times S^5$ are the fieds on the (anti)-de Sitter space. There are even more percularities in AdS as compared to the flat Minkowski. – Eugene Starling Feb 27 '10 at 08:29
  • As for Minkowski, the veilbein and the spin connection can be recognized together as the single connection of the Poincare algebra, which is the symmetry algebra of the Minkowski space. In physics notation $A=e^a P_a+\omega^{a,b} M_{ab}$, where $P_a$, $M_{ab}$ are the generators of the Poincare algebra satisfying $[P_a,P_b]=0$, $[P_a,M_{bc}]=\eta_{ab}P_c-...$, $[M_{ab},M_{cd}]=\eta_{ac}M_{bd}+...$ The covariant derivative $d+A$ may act on any bundle, tensor or spin-tensor. – Eugene Starling Feb 27 '10 at 08:31
  • Thanks Eugene. The physics notation of A that you mentioned is the quantity whose basis free definition I am trying to understand.

    Further there is also this issue that say on $S^5$ which can be thought of as $SO(5)/SO(4)$ aren't the spins of particles defined through irreducible representations of $SO(4)$ instead of $SU(2)$ ?

    Things go back to the initial question that on a given manifold what is the definition of spin?

    Further if we keep to $SU(2)$ spins what is the importance of transverse traceless tensors in understanding spins? Is there some map between the two?

     – Anirbit Feb 27 '10 at 12:54
  • @Anirbit: the five-sphere is $SO(6)/SO(5)$. – José Figueroa-O'Farrill Feb 27 '10 at 12:59
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    @Anirbit: homogeneous vector bundles on a homogeneous space $G/H$ are in one-to-one correspondence with representations of $H$. Minkowski spacetime is a homogeneous space of the Poincaré group $P$ with isotropy isomorphic to the Lorentz group $L$, hence we can view it as $P/L$ and hence homogeneous vector bundles are in one-to-one correspondence with representations of $L$. However the construction of the unitary representations of the Poincaré group takes place in "momentum space" and then gets Fourier-transformed to Minkowski spacetime. (tbc) – José Figueroa-O'Farrill Feb 27 '10 at 13:03
  • (cont'd) This should be explained in Weinberg's Volume 1, Chapter 2. As for the connection $A$ in Eugene's comment, this is an example of a Cartan connection, which is a different object than the spin connection. You can read about this in Sharpe's book on Differential Geometry or perhaps in this eprint of Derek Wise: http://arxiv.org/abs/gr-qc/0611154 – José Figueroa-O'Farrill Feb 27 '10 at 13:06
  • Ah..I should have said $S^4$ . The question goes for any $S^n=SO(n+1)/SO(n)$ :)

    Thanks for the explanation about why the fields should be sections of a homogeneous bundle on the Minkowski space. And I suppose the spin determines the rank of this bundle to be 2s+1 ? But is it true that any space-time can be thought of as you said? And I suppose we also mean that only its spin degree of freedom lives in this homogeneous bundle?

     – Anirbit Feb 27 '10 at 13:39
  • Yes, the rank of the bundle is the dimension of the representation of $H$ (in $G/H$). Now this is only true for homogeneous spacetimes. You can talk about classical fields in any spacetime as sections of vector (or more generally fibre) bundles associated to the bundle of frames or, in the case of spin manifolds, to the spin bundle. These are associated to representations of the corresponding orthogonal or spin group. (tbc) – José Figueroa-O'Farrill Feb 27 '10 at 14:52
  • (cont'd) If by spacetime you mean an $n$-dimensional lorentzian spin manifold --- that is, signature $(n-1,1)$ --- then classical fields are associated to representations of $\mathrm{Spin}(n-1,1)$. If and only if $-1$ acts trivially, do these representations factor through $\mathrm{SO}(n-1,1)$ in which case they "tensorial" representations. – José Figueroa-O'Farrill Feb 27 '10 at 14:56