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Qmechanic ♦
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I am reading Schwartz's QFT and am a little lost in ch 8.2 in which we are trying to "embed particles into fields".

I think I am missing some very (obvious?) high level intuition for what is going on here.

What I understand so far is that we want to find a set of basis states for a given particle that form a representation of the Poincare group. In particular this basis should be irreducible (this means that there is no subset of the states that are also representations) and unitary (this means that the matrix elements $\mathcal{M}$ are invariant under transformations.)

There is a brief mention of a proof by Wigner that the only unitary irreducible representations of the Poincaré group are infinite. They can be classified by $m$, which is continuous  ($m^2=p^2$) and J$J$ (the spin we know from quantum mechanics). Further, given an allowed value for $J$ there are $2J+1$ states for a given mass.

Everything feels good up to this point until Schwartz states that, "For spin 0, the embedding is easy, we just put the one degree of freedom into a J=0$J=0$ scalar field." HowHow is this "easy", and where did the one degree of freedom go? IsIs this just a statement that the field is a scalar and not a tensor? Could we equally well embed J=0$J=0$ in a degenerate tensor $a_\mu =(a,a,a,a) $, for example?

I am reading Schwartz's QFT and am a little lost in ch 8.2 in which we are trying to "embed particles into fields".

I think I am missing some very (obvious?) high level intuition for what is going on here.

What I understand so far is that we want to find a set of basis states for a given particle that form a representation of the Poincare group. In particular this basis should be irreducible (this means that there is no subset of the states that are also representations) and unitary (this means that the matrix elements $\mathcal{M}$ are invariant under transformations.)

There is a brief mention of a proof by Wigner that the only unitary irreducible representations of the Poincaré group are infinite. They can be classified by $m$, which is continuous($m^2=p^2$) and J (the spin we know from quantum mechanics). Further, given an allowed value for $J$ there are $2J+1$ states for a given mass.

Everything feels good up to this point until Schwartz states that, "For spin 0, the embedding is easy, we just put the one degree of freedom into a J=0 scalar field." How is this "easy", and where did the one degree of freedom go? Is this just a statement that the field is a scalar and not a tensor? Could we equally well embed J=0 in a degenerate tensor $a_\mu =(a,a,a,a) $, for example?

I am reading Schwartz's QFT and am a little lost in ch 8.2 in which we are trying to "embed particles into fields".

I think I am missing some very (obvious?) high level intuition for what is going on here.

What I understand so far is that we want to find a set of basis states for a given particle that form a representation of the Poincare group. In particular this basis should be irreducible (this means that there is no subset of the states that are also representations) and unitary (this means that the matrix elements $\mathcal{M}$ are invariant under transformations.)

There is a brief mention of a proof by Wigner that the only unitary irreducible representations of the Poincaré group are infinite. They can be classified by $m$, which is continuous  ($m^2=p^2$) and $J$ (the spin we know from quantum mechanics). Further, given an allowed value for $J$ there are $2J+1$ states for a given mass.

Everything feels good up to this point until Schwartz states that, "For spin 0, the embedding is easy, we just put the one degree of freedom into a $J=0$ scalar field." How is this "easy", and where did the one degree of freedom go? Is this just a statement that the field is a scalar and not a tensor? Could we equally well embed $J=0$ in a degenerate tensor $a_\mu =(a,a,a,a) $, for example?

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Embedding particles (with spin) into fields (Schwartz qft)

I am reading Schwartz's QFT and am a little lost in ch 8.2 in which we are trying to "embed particles into fields".

I think I am missing some very (obvious?) high level intuition for what is going on here.

What I understand so far is that we want to find a set of basis states for a given particle that form a representation of the Poincare group. In particular this basis should be irreducible (this means that there is no subset of the states that are also representations) and unitary (this means that the matrix elements $\mathcal{M}$ are invariant under transformations.)

There is a brief mention of a proof by Wigner that the only unitary irreducible representations of the Poincaré group are infinite. They can be classified by $m$, which is continuous($m^2=p^2$) and J (the spin we know from quantum mechanics). Further, given an allowed value for $J$ there are $2J+1$ states for a given mass.

Everything feels good up to this point until Schwartz states that, "For spin 0, the embedding is easy, we just put the one degree of freedom into a J=0 scalar field." How is this "easy", and where did the one degree of freedom go? Is this just a statement that the field is a scalar and not a tensor? Could we equally well embed J=0 in a degenerate tensor $a_\mu =(a,a,a,a) $, for example?