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Some actions like 11D supergravity are defined "on-shell". What does this mean exactly? Can you give me an example? Say for example the Klein-Gordon action. Can this be defined on-shell too?

Qmechanic
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  • Also have a look at my answer here: http://physics.stackexchange.com/q/108546/ – JamalS Nov 04 '15 at 13:19
  • I've never heard this expression, can you give an example (a book or so) where it is used? Normally, "on-shell" means that the equations of motion have been used to simplify a term. But if you plug the e.o.m. into the action, you can not use the action any more to derive the e.o.m. ... – Noiralef Nov 04 '15 at 18:18
  • Here is one: http://www.sciencedirect.com/science/article/pii/0550321381903941 I think you are right. Maybe it means that you need the equations of motion for the supersymmetry to close? Perhaps this means it is only valid classically not quantum mechanically? I'm not sure. –  Nov 04 '15 at 18:28

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The fields of a supersymmetric theory form a representation of the super Poincare algebra. When this representation is restricted to a specific value of the mass operator $P^{\mu}P_{\mu} = m^2$, the representation is called an on shell representation multiplet.

On shell representations are characterized by the equality of the number of bosonic and fermionic states. When we try to construct supersymmetric Lagrangians based on the fields from the on shell representation multiplets, we observe that the algebra of the super Poincare Noether charges closes only for field configurations satisfying the equation of motion. This is the reason why such actions are called on-shell.

The deeper reason to this problem is that in contrast to the states, the bosonic and fermionic fields (according to their spin) have numbers of components differing from the corresponding numbers of states (except for the neutral scalar field). But supersymmetry has a strict rule of Boson # = Fermion # at any level, so there is a need of auxiliary bosonic fields to balance the fermionic number of components. There are cases when the additional (auxiliary) fields are can be appropriately added to the Lagrangian such that the supersymmetry algebra based on the full set of fields closes without the application of the equations of motion. In this case, the multiplet composed of the original and the auxiliary fields forms an off-shell representation of the supersymmetry algebra. The equations of motion of these auxiliary fields are algebraic, thus they are not dynamical and whose degrees of freedom vanish on-shell.

There are supersymmetric theories which do not have known off-shell formulations. May be such formulations do not even exist. This includes all supersymmetric theories in dimension larger than 6. On the other hand the minimal supergravity theory in 4 dimensions has may inequivalent off-shell formulations.

The reason that an off-shell formulation is believed to be necessary is that the on-shell formulation is not suitable for a path integral quantization. While the trivial path integration of the auxiliary fields can be performed to obtain the on-shell theory, the nonclosure of the supersymmetry algebra must be imposed as a constraint in the quantum formulation in order to obtain a fully supersymmetric quantum theory. These constraint surfaces are complicated making the quantization problem highly non-trivial. There are several techniques which allow the construction of the off-shell action given the on-shell action (whenever it is possible), most of which are based on superspace formulation.

The following references contain more elaboration on this subject:

1) Please see the following review by Sohnius especially section 5.4 treating this subject.

2) The following review by Gates, Linch, Philips and Rana, describes the relatively state of the art open problems of off shell formulation of supersymmetric theories.

3) Also, the following relatively recent research statement by Gregory Landweber and the references therein describes some advanced techniques that people are trying to use for the construction of off-shell supersymmetric actions.

David Bar Moshe
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    Thank you that is a brilliant answer! So does that mean that a super-symmetric theory which is defined only on-shell can't be quantised? Is string theory off-shell or on-shell? –  Nov 07 '15 at 22:09
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    When the supersymmetry algebra does not close of-shell, then supersymmetry will not be preserved in the quantum theory. Thus even if the theory can be quantized, it would not be supersymmetric at the quantum level. The situation in supersymmetric gauge theories is even more complicated, because there are cases where the closure of the BRST algebra requires the closure of supersymmetry algebra. Strings viewed as 2 dimensional sigma models can be quantized while preserving the world sheet supersymmetry (It is not technically easy in the Neveu-Schwarz case). ... – David Bar Moshe Nov 11 '15 at 13:35
  • ... However there are difficulties in attaining the space time supersymmetry at the quantum level. One major recent advancement in this direction is the introduction of pure spinor Green-Schwarz superstrings. – David Bar Moshe Nov 11 '15 at 13:35