Is the ground state energy of a quantum field actually zero?

[vanhees71]

Sigh, I think this is really a superfluous discussion.

If the proper orthochronous Poincare group in the classical sense was the very group you have to use in QT, which you must if you insist on that only unitary representations of the symmetry groups of physics are "allowed descriptions" of symmetry principles in QT, then you'd not be allowed to use the covering group of the rotation group (as a subgroup of the Poincare group) and then only integer-spin representations would be allowed. As observations in Nature, however, show there are half-integer spin realizations of the group in nature like electrons, nucleons, etc. etc.

 

[PeterDonis]

particles with half integer spin are represented in QFT already by a vector representation of the Poincare group

Don't you mean a spinor representation? A vector representation would be spin 1, not spin 1/2, correct?

 

[A. Neumaier]

No; you mix conceptually completely different things.

Weinberg's argument allows arbitrary shifts of the angular momentum in an unphysical central extension.

On the other hand, particles with half integer spin are represented in QFT already by a vector representation of the Poincare group, no central extension is necessary to do so!

Sigh, I think this is really a superfluous discussion.

If the proper orthochronous Poincare group in the classical sense was the very group you have to use in QT, which you must if you insist on that only unitary representations of the symmetry groups of physics are "allowed descriptions" of symmetry principles in QT, then you'd not be allowed to use the covering group of the rotation group (as a subgroup of the Poincare group) and then only integer-spin representations would be allowed. As observations in Nature, however, show there are half-integer spin realizations of the group in nature like electrons, nucleons, etc. etc.

Sorry, of course one has the phases in the unitary transformations that give a central extension of order 2, i.e., one has a vector representation of ISL(2,C) rather than one of ISO(1,3). Note that these have the same Lie algebra commutation relations defining the standard generators without any shift! You should therefore interpret my comments to apply to ISL(2,C) rather than the Poincare group.

Weinberg posits an arbitrary central extension of this Lie algebra (thus changing the definition and hence the meaning of the generators by positing different commutation rules) and shows that no physics can result, hence that this Lie algebra extension is physically spurious - in contrast to the Galilei group where a nontrivial central extension with mass as a central charge (the one realized in nonrelativistic physics) appears by the same kind of analysis.

In general, physical observables with a meaning in terms of symmetry are (in all cases, without exception) defined on the theoretical level by their commutation rule. If the commutation rule change, the meaning of the observables change. In the extended Lie algebra considered by Weinberg, all generators, including those for the rotation group, alter their meaning by being shifted.

You can apply exactly the same argument to SO(3) or SU(2) - which one doesn't matter since their Lie algebra is the same and Weinberg only argues with the Lie algebra. After similar calculations you end up with the same result - that there is no central charge. So if one follows your argument one should conclude that angular momentum is physically determined only up to an arbitrary shift in each component. Of course, this is nonsense, hence your argument implies no such thing - also not in the case of energy where you originally applied it.

This is completely independent of the double-valuedness of spin 1/2. Ths enters the discussion only on the group level, but Weinberg's argument is solely on the Lie algebra level! Therefore the conclusions also apply only on the Lie algebra level! This is what i had meant when saying that you mix two completely different things!

 

[A. Neumaier]

Don't you mean a spinor representation? A vector representation would be spin 1, not spin 1/2, correct?

Yes. This is equivalent to a vector representation of SU(2), which has the same Lie algebra as SO(3), generated by the components of angular momentum, without involving a central extension on the Lie algebra level. The situation discussed earlier is the same except that the group ISL(2,C) is bigger, but it too has the same Lie algebra as the Poincare algebra ISO(1,3), also without involving a central extension on the Lie algebra level.

Thus the phenomenon of half integer spin has nothing at all to do with a discussion such as Weinberg's or vanhees71's, where the Lie algebra is augmented by a (in both cases spurious) central charge.

 

[PeterDonis]

This is equivalent to a vector representation of SU(2),

Probably this is just an issue of terminology. By "vector representation of SU(2)", do you mean the fundamental representation? As described, for example, here:

https://en.wikipedia.org/wiki/Representation_theory_of_SU(2)

 

[A. Neumaier]

"vector representation of SU(2)", do you mean the fundamental representation?

No. All representations of SU(2) are vector representations, i.e., representations on a vector space - in contrast to ray representations, which are representations on a projective space.

 

[PeterDonis]

vector representations, i.e., representations on a vector space

Ah, ok, as I thought, you are using the term "vector representation" differently from the way I'm used to seeing it used. Your usage is much more general.

 

[A. Neumaier]

Ah, ok, as I thought, you are using the term "vector representation" differently from the way I'm used to seeing it used. Your usage is much more general.

"vector representation" = "linear representation" is analogous to
"ray representation" = "projective representation"; cf. https://en.wikipedia.org/wiki/Projective_representation

One can either put the emphasis on the elements on which the group acts, or on the whole collection of such elements.

 

[samalkhaiat]

Sigh, I think this is really a superfluous discussion.

If the proper orthochronous Poincare group in the classical sense was the very group you have to use in QT, which you must if you insist on that only unitary representations of the symmetry groups of physics are "allowed descriptions" of symmetry principles in QT, then you'd not be allowed to use the covering group of the rotation group (as a subgroup of the Poincare group) and then only integer-spin representations would be allowed. As observations in Nature, however, show there are half-integer spin realizations of the group in nature like electrons, nucleons, etc. etc.

You are absolutely correct. You cannot obtain spinors from the (classical) symmetry groups $\mbox{SO}(3)$ and $\mbox{SO}(1,3)$.