why cannot fermions have non-zero vacuum expectation value?

Question

In quantum field theory, scalar can take non-zero vacuum expectation value(vev). And this way they break symmetry of the Lagrangian. Now my question is what will happen if the fermions in the theory take non-zero vacuum expectation value? What forbids fermions to take vevs?

This post imported from StackExchange Physics at 2014-04-13 11:31 (UCT), posted by SE-user Paul

Comments

The Higgs has a non-zero vev, but there's no reason to suppose this applies to other scalars. It's a special property of the Higgs.

This post imported from StackExchange Physics at 2014-04-13 11:31 (UCT), posted by SE-user John Rennie

Answer 1

Why can't fermions have a non-zero vacuum expectation value (VEV)? Lorentz invariance.

If anything other than a Lorentz scalar has a non-zero VEV, Lorentz invariance would be spontaneously broken.

For example, suppose we have a Lorentz invariant term in a Lagrangian for a vector $$ \mathcal{L} \supset m^2 A_\mu A^\mu. $$ Now suppose the vector obtains a VEV, $A_\mu \to v + A_\mu$, $$ m^2 A_\mu A^\mu \to m^2 v A^\mu + m^2 vA_\mu + m^2v^2 + m^2 A_\mu A^\mu. $$ The first two are clearly not Lorentz invariant. One can construct idential arguments for any non-scalar field term. If $\psi\to v+\psi$, the VEV, $v$, won't have the same Lorentz transformation properties as the field, $\psi$ unless $\psi$ is a scalar.

This post imported from StackExchange Physics at 2014-04-13 11:31 (UCT), posted by SE-user innisfree

Comments

Hi @innisfree, thanks for your answer. But, it will be very helpful, if you could explain it a bit more mathematically?

This post imported from StackExchange Physics at 2014-04-13 11:31 (UCT), posted by SE-user Paul

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Take the fermion field to have spin 1/2. A spin 1/2 defines a direction in space, the direction such that the spin 1/2 is "up" with respect to that direction. If this field interacts with other fields, this breaks rotational symmetry. The only spin that does not define a preferred direction in this way is spin 0, that is, a scalar.

This post imported from StackExchange Physics at 2014-04-13 11:31 (UCT), posted by SE-user Robin Ekman

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@Paul, is it clearer now?

This post imported from StackExchange Physics at 2014-04-13 11:31 (UCT), posted by SE-user innisfree

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Having said that, I suppose there could be very special theories, in which e.g. vectors appeared always as $\partial_\mu A^\mu$, allowing a VEV.

This post imported from StackExchange Physics at 2014-04-13 11:31 (UCT), posted by SE-user innisfree

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@innisfree: I don't think this has to do with Lorentz invariance, but that's it is a general fact about Grassmanian fields, though I don't know the proof...

This post imported from StackExchange Physics at 2014-04-13 11:31 (UCT), posted by SE-user Adam

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@innisfree: In any "derivatively-coupled" theory, one cannot have a vev, right? Since there is no special point in field space.

This post imported from StackExchange Physics at 2014-04-13 11:31 (UCT), posted by SE-user Siva

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@Paul: VEV = Vacuum expectation value i.e. a property of the vacuum. So if any object in a non-trivial representation of the Poincare algebra picks up a VEV, then some of the spacetime symmetries will be spontaneously broken by the vacuum (state). One can expect the same to apply to fluctuations around the vacuum. That would mean that the corresponding conserved quantities are not really conserved. And as far as we can see, conservation of energy-momentum and angular momentum apply quite perfectly to our universe.

This post imported from StackExchange Physics at 2014-04-13 11:31 (UCT), posted by SE-user Siva

Answer 2

I think it is a general fact about grassmannian field, and this has nothing to do with Lorentz invariance or other symmetries (you can invent a lot of QFTs without this kind of symmetry, but the VEV of a fermionic operator will be always zero (in the absence of sources)).

In a functional integral formulation, the VEV of a grassmannian field $\psi$ is written as $$ \langle \psi \rangle= \int D\psi D\bar\psi\, \psi \,e^{-S},$$ where the action S is bosonic (involves products even products of $\psi$ and $\bar\psi$). Therefore, unless there are source terms of the form $\bar\eta\psi$ in the action, the integral over the $\psi e^{-S}$ will give zero, since we are integrating over an odd number of grassmannian fields (when the exponential is expanded).

This post imported from StackExchange Physics at 2014-04-13 11:31 (UCT), posted by SE-user Adam

Comments

Can't one make similar arguments for a scalar vev $<\phi>=0$? It ought to be zero, which is why we expand about the homogeneous nonzero part in Higgs mechanisn

This post imported from StackExchange Physics at 2014-04-13 11:31 (UCT), posted by SE-user innisfree

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Any field linear in creation and annihilation operators will give a zero vev

This post imported from StackExchange Physics at 2014-04-13 11:31 (UCT), posted by SE-user innisfree

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@innisfree: Not necessarily. If the action is not symmetric (say to $\phi\to-\phi$, where $\phi$ is bosonic), then you can have $\langle\phi\rangle\neq 0$. For creation operator, this is not the case, if you include a source $J$, compute $\langle\hat a^\dagger\rangle$ at finite source, and the let $J\to0$ (and if there is a degeneracy between two states with a difference of one particle). For fermions, you will always find that as $\eta\to0$, then $\langle\psi\rangle\to0$.

This post imported from StackExchange Physics at 2014-04-13 11:31 (UCT), posted by SE-user Adam

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_Any_ expectation value of a fermion field vanishes, not only the VEV.

Answer 3

Well, they could develop a vev, depending of the interactions you consider. Take e.g. this term in the lagrangian $L=\frac{1}{\Lambda^2}[\bar{\psi}\psi- v^3]^2$. Of course this would imply that you also break Lorentz, which you may want to avoid.

This post imported from StackExchange Physics at 2014-04-13 11:31 (UCT), posted by SE-user TwoBs