If a symmetry operator S in a QFT annihilates the vacuum, why does S preserve the space of 1-particle states?

Question

In the paper "Supersymmetry and Morse Theory", on the third page (p. 663 in the journal version), Witten says:

"Now in any quantum field theory if a symmetry operator (an operator which commutes with the Hamiltonian) annihilates the vacuum state, then the one particle states furnish a representation of the symmetry."

Why is this true? Is there a simple explanation or computation that doesn't go too far afield of Witten's relatively informal discussion in the introduction of this paper, or is it more complicated?

This post imported from StackExchange Physics at 2015-05-09 14:34 (UTC), posted by SE-user physics_acolyte

Comments

Well, from classical quantum mechanics it is somehow self-evident why are excitations of the vacuum meaningfully organized by representations of the respective symmetries - all the classical problems of QM from the infinite potential well to the hydrogen atom can be understood as representation-theory problems. However, the hard question of QFT (as far as my limited knowledge reaches, unanswered) is why are particles "renormalized irreducible quadratic excitations". So far, I have understood this as a postulate of QFT rather than some kind of mathematical necessity.

This post imported from StackExchange Physics at 2015-05-09 14:34 (UTC), posted by SE-user Void

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What does he mean by "the one particle states furnish a representation of the symmetry." Is he basically saying that the symmetry maps one particle states into one particle states?

This post imported from StackExchange Physics at 2015-05-09 14:34 (UTC), posted by SE-user NowIGetToLearnWhatAHeadIs

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The case $\hat{H}\propto\hat{n}$ is trivial. $\hat{H}\hat{S}|n\rangle = \hat{S}\hat{H}|n\rangle = n\hat{S}|n\rangle$, so $\hat{S}|n\rangle$ has the same number of particles as $|n\rangle$. In other words, $\hat{S}$ preserves particle number. A "free" field theory is precisely one in which $\hat{H}$ is a sum of $\hat{n}$ operators for every possible mode, so we've shown that the statement is true for a free theory. For an interacting theory I don't know.

This post imported from StackExchange Physics at 2015-06-16 14:50 (UTC), posted by SE-user DanielSank

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More answers (among them mine) together with an extensive discussion can be found at physicsoverflow.org/30822

This post imported from StackExchange Physics at 2015-06-16 14:50 (UTC), posted by SE-user Arnold Neumaier

Answer 1

This is a heuristic explanation of Witten's statement, without going into the subtleties of axiomatic quantum field theory issues, such as vacuum polarization or renormalization.

A particle is characterized by a definite momentum plus possible other quantum numbers. Thus, one particle states are by definition states with a definite eigenvalues of the momentum operator, they can have further quantum numbers. These states should exist even in an interactiong field theory, describing a single particle away from any interaction. In a local quantum field theory, these states are associated with local field operators: $$| p, \sigma \rangle = \int e^{ipx} \psi_{\sigma}^{\dagger}(x) |0\rangle d^4x$$ Where $\psi $ is the field corresponding to the particle and $\sigma$ describes the set of other quantum numbers additional to the momentum. A symmetry generator $Q$ being the integral of a charge density according to the Noether's theorem $$Q = \int j_0(x') d^3x'$$ should generate a local field when it acts on a local field: $[Q, \psi_1(x)] = \psi_2(x)$ (In the case of internal symmetries $\psi_2$ depends linearly on the components of $\psi_1(x)$, in the case of space time symmetries it depends on the derivatives of the components of $\psi_1(x)$)

Thus in general:

$$[Q, \psi_{\sigma}(x)] = \sum_{\sigma'} C_{\sigma\sigma'}(i\nabla)\psi_{\sigma'}(x)])$$

Where the dependence of the coefficients $ C_{\sigma\sigma'}$ on the momentum operator $\nabla$ is due to the possibility that $Q$ contains a space-time symmetry. Thus for an operator $Q$ satisfying $Q|0\rangle = 0$, we have $$ Q | p, \sigma \rangle = \int e^{ipx} Q \psi_{\sigma}^{\dagger}(x) |0\rangle d^4x = \int e^{ipx} [Q , \psi_{\sigma}^{\dagger}(x)] |0\rangle d^4x = \int e^{ipx} \sum_{\sigma'} C_{\sigma\sigma'}(i\nabla)\psi_{\sigma'}(x) |0\rangle d^4x = \sum_{\sigma'} C_{\sigma\sigma'}(p) \int e^{ipx} \psi_{\sigma'}^{\dagger}(x) |0\rangle d^4x = \sum_{\sigma'} C_{\sigma\sigma'}(p) | p, \sigma' \rangle $$ Thus the action of the operator $Q$ is a representation in the one particle states. The fact that $Q$ commutes with the Hamiltonian is responsible for the energy degeneracy of its action, i.e., the states $| p, \sigma \rangle$ and $Q| p, \sigma \rangle$ have the same energy.

This post imported from StackExchange Physics at 2015-06-16 14:50 (UTC), posted by SE-user David Bar Moshe

Comments

1) Why does $[Q,\psi]$ have to produce a strictly local field? I guess that the symmetry of the Hamiltonian may lead to the requirement that it does but I do not see it. 2) Do indeed the statements hold in interacting field theory? Could you maybe point to a reference where it is shown that the fullblown self-interacting one-particle states also furnish a representation of the symmetry?

This post imported from StackExchange Physics at 2015-06-16 14:51 (UTC), posted by SE-user Void

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@Void 1) Please think for a moment of $Q$ as the electric charge operator, you can write it uzing the Gauss' law as a surface integral of the electric field over a very large sphere at infinity. Due to the large distance these fields do not produce singularities when multiplied by other fields, thus the only singualrities coming from the commutator are those due to the local field $\psi$, thus the commutator itself is also a local field at $x$.

This post imported from StackExchange Physics at 2015-06-16 14:51 (UTC), posted by SE-user David Bar Moshe

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@Void 2) Assuming Lorentz symmetry, the renormalization factors Z in ψ R =Z(ψ)ψ are Lorentz scalars, thus nothing essential changes in the analysis when the true renormalized field is used: Its transformation properties remain the same.

This post imported from StackExchange Physics at 2015-06-16 14:51 (UTC), posted by SE-user David Bar Moshe

Answer 2

This is a conceptual issue; only trivial computations are needed. Note that for an interacting theory, the 1-particle states are the (renormalized) states of the interacting theory and not the(bare) states of the free theory in terms of which the bare Hamiltonian is formulated!

The vacuum states of a Poincare covariant quantum field theory are defined as the states of the trivial (1-dimensional) representation of the Poincare group in the decomposition of the unitary representation of the Poincare group on the vacuum sector of the observable algebra of the theory. Similarly, the 1-particle states of the theory are defined as the states of the irreducible representations of the Poincare group in this decomposition. 

If a self-adjoint operator $Q$ generates a 1-parameter group of symmetries $U(s):=e^{isQ}$ of the theory, it commutes with all generators of the Poincare group, i.e., if for each Poincare transformation $\Lambda$ and all real $s$,, the transform $U(s)\Lambda U(-s)$ is another Poincare transformation. This implies that the image of a representation space of the Poincare group under $U(s)$ is another representation space of the Poincare group. In particular, since symmetries are by definition invertible, the image of a vacuum state under a symmetry $U(s)$ has dimension $\le 1$, hence must be another vacuum state, and the image of an irreducible 1-particle space under a symmetry $U(s)$ is again irreducible, hence another irreducible 1-particle space. In particular, the space spanned by all vacuum states and the space spanned by all 1-particle states are both invariant under the symmetry, and hence furnishes a representation of it. (In particular, the assumption that $Q$ annihilates a vacuum state is not needed for this argument.)

Comments

@ArnoldNeumaier; Your argument really only works when Q annihilates the vacuum, because only in this case is exp(iQ) well defined. The assumption of an invariant vacuum is needed because when the symmetry is spontaneously broken, exp(iQ) on a single particle state symmetry rotates both the particle and the whole vacuum, so in addition to transforming the particle, it produces infinitely many 0-momentum Goldstone bosons, so it isn't even defined as an operator in the usual Hilbert space.

If you formally expand the Hilbert space in a nonseparable way to include all the different vacua in a single Hibert space which is not physical (nobody does this in a canonical formalism, but this is what the path integral does in practice), then it is true that the image of exp(iQ) where Q is the broken symmetry, is still a single particle state, for the reason you say, but it's a single particle state in a completely different vacuum.

@JiaYiyang: The argument is talking about an internal symmetry, just ignore supersymmetries and the Poincare group generators.

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 I just don't see how invertiblity implies preservation of irreduciblity. So is "symmetry must preserve irreduciblity" part of your definition of a symmetry?

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@JiaYiYang: If the image representation were reducible you could split it into at least to, and take the preimages to get reducibility of the original representation.

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@ArnoldNeumaier, 

Yeah you are right, +1 to your answer. But I still wonder if you adopt Witten's definition of symmetry (an operator which commutes with the Hamiltonian), will you still be able to get the same result?

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@JiaYiYang: I don't think so. His definition is in any case flawed; I don't think that he meant it literally.

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A valuable discussion about the question "Is the path integral in 4 dimensions still only at the level of useful heuristic?" that is however off-topic to the question at hand, has been moved here to chat.

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In particular, since symmetries are by definition invertible, the image of a vacuum state under a symmetry U(s) is another vacuum state, and the image of an irreducible 1-particle space under a symmetry U(s) is another irreducible 1-particle space.

I don't quite mind your conclusion on the vacuum state, since it's implied by the assumption of $Q$ annihilating the vacuum anyway. However, it still seems like a non sequitur to me that an invertible $U$ should map 1-particle states to 1-particle states, or in other words preserving the irreduciblity. 

Answer 3

@NowIGetToLearnWhatAHeadIs said in the comment 

What does he mean by "the one particle states furnish a representation of the symmetry." Is he basically saying that the symmetry maps one particle states into one particle states?
 

I think this is basically it. To prove it, you need to show the Hermitian generator $Q$ of your symmetry has the properties:

(1)$\langle 0|Q|1\rangle=0$,

(2)$\langle n|Q|1\rangle=0$ for $n>1$.

(1) has to be true since $Q$ is Hermitian and annihilates vacuum, and you just let $Q$ act to the left. For (2) I can't say I have a completely rigorous proof, but here's my heuristics: For $\langle n|Q|1\rangle$ to be nonzero, $Q$ must increase the particle number when acting on $|1\rangle$, and such an operator must also increase the particle number when acting on vacuum(this is where I don't I have a rigorous proof), and if so we arrive at a contradiction, and we conclude (2) must be true.

Note that reasoning for (2) is far from complete.

Comments

$Q=a^*a^*a$ annihilates the vacuum but creates a 2-particle state from a 1-particle state; so the argument must be more involved.

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@ArnoldNeumaier, yes, I'm suspecting Hermiticity might play a role.

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Yes. To turn this into a complete argument, you'd have to assume that you can expand $Q$ into a series in normally ordered c/a operators, then a hermiticity argument applies and gives after some more work the desired contradiction. But the whole argument works only under the assumption that the interacting theory is in a Fock representation, and even then it is not fully rigorous.

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@JiaYiyang; The way you prove the result is to fold in the proof of Goldstone's theorem. You need to know that $Q = \int j_0(x) d^3x$, there there is a local current for the symmetry, and then consider smoothed out finite volume versions of Q $\int j_0(x) \rho(x) d^3x$. When the symmetry is spontaneously broken, the current produces a Goldstone boson, otherwise it just rotates the one particle states into symmetry related one particle states. This is standard current algebra, Gell-Mann's paper on this is really good.

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@RonMaimon, thanks, let me think about it.

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@ArnoldNeumaier, actually $Q=a^*a^*a+a^*aa$ is a valid counterexample. I guess I'll try to think along the line Ron suggested.

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@RonMaimon, Could you give a specific reference of Gell-Mann's paper you mentioned? 

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@JiaYiyang: That paper will be useless, I should delete the comment. I wanted to write an answer, but there is one special case which is interesting to consider because it might include a nonlocal counterexample--- a massless particle of energy E can be related by symmetry to two massless particles of energy E/2, this isn't fobidden by energy conservation. You can imagine making a representation of a symmetry operation which mixed up massless particles of one type with pairs of massless particles of another type (or triplets, quadruplets, etc). The point is to do it so that any type of particle can be symmetry related only to different number of other particles. I can't rule this out, it might be possible using a nonlocal symmetry, but I can't think of any example either. Witten's statement might have to be stated carefully about a symmetry with a local current, rather than just something which commutes with the Hamiltonian. Not sure. Forget about the Gell-Mann paper, I was just being lazy.

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@RonMaimon, @ArnoldNeumaier,  this is a highly incomplete answer, do you mind if I hide my answer?

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 The discussion is not completely without value, but if you still want to hide it, I wouldn't mind.

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@JiaYiyang: I can't tell you what to do, but don't worry, you aren't embarrassing yourself. Making statements like this that are intuitively obvious rigorous is annoying, and depends on the assumptions you take. I just want to understand whether it is possible for massless particles to have symmetries that relate different numbers of particles moving the same way to each other. I can't think of any way to do it, but that's probably my limited imagination.

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@ArnoldNeumaier, @RonMaimon, ok, then I'll let it stay.