I'm quite confused by two short paragraphs in Schwarz 28. He proves that $$ Q = \int d^3 x J_0(x) = \int d^3 x \sum_m \frac{\partial L}{\partial \dot \phi_m} \frac{\delta \phi_m}{\delta \alpha} $$ (where $ J_\mu $ is a conserved current) is a generator for the symmetry transformation, which is fine, and that $ Q | \Omega\rangle $ is degenerate with the ground state due to $H$ commuting with $Q$ due to charge conservation.

Fine, the issue is when he states $$ | \pi(\bar p ) \rangle = \frac{-2i}{F} \int d^3 x exp(i \bar p \cdot \bar x ) J_0(x) | \Omega \rangle .$$ If I insert one of the definitions above into the below, we end up with something like $$ \int d^3 x exp(i \bar p \cdot \bar x ) \sum_m \pi_m(x) \frac{d\phi_m}{d\alpha} .$$

1. Is this just a definition, or is there a deeper mathematical reason? Intuitively, it makes sense since we expect $ p = 0 $ to produce $ | \pi(0 ) \rangle$ so it's just like we're moving it in momentum space.

2. Furthermore, he then states they have energy $ E(p) + E_0 $, which also confuses me: when applying $H$ to the above, assuming we can pass it into the integral, I'd expect to see some kind of obvious separation like $$ H( \int d^3 x J_0 (x) | \Omega\rangle ) + H (something else) $$ to correspond to the form above, but this is also unclear to me on how to proceed.

3. I'm also confused on his final assertion, that since $ E(p) \rightarrow 0$ as $ p \rightarrow 0 $, (which makes sense), that there is a massless dispersion relation. this means that $E(k) $ is not dependent on mass. this isn't obvious to me, since if it is of the form $ m k $ or something silly of this nature, then it will still drop out as $ k \rightarrow 0$.

I'm quite confused by two short paragraphs in Schwarz 28. He proves that $$ Q = \int d^3 x J_0(x) = \int d^3 x \sum_m \frac{\partial L}{\partial \dot \phi_m} \frac{\delta \phi_m}{\delta \alpha} \tag{28.5}$$ (where $ J_\mu $ is a conserved current) is a generator for the symmetry transformation, which is fine, and that $ Q | \Omega\rangle $ is degenerate with the ground state due to $H$ commuting with $Q$ due to charge conservation.

Fine, the issue is when he states $$ | \pi(\bar p ) \rangle = \frac{-2i}{F} \int d^3 x \exp(i \bar p \cdot \bar x ) J_0(x) | \Omega \rangle .\tag{28.8}$$ If I insert one of the definitions above into the below, we end up with something like $$ \int d^3 x \exp(i \bar p \cdot \bar x ) \sum_m \pi_m(x) \frac{d\phi_m}{d\alpha} .$$

1. Is this just a definition, or is there a deeper mathematical reason? Intuitively, it makes sense since we expect $ p = 0 $ to produce $ | \pi(0 ) \rangle$ so it's just like we're moving it in momentum space.

2. Furthermore, he then states they have energy $ E(p) + E_0 $, which also confuses me: when applying $H$ to the above, assuming we can pass it into the integral, I'd expect to see some kind of obvious separation like $$ H( \int d^3 x J_0 (x) | \Omega\rangle ) + H (something else) $$ to correspond to the form above, but this is also unclear to me on how to proceed.

3. I'm also confused on his final assertion, that since $ E(p) \rightarrow 0$ as $ p \rightarrow 0 $, (which makes sense), that there is a massless dispersion relation. this means that $E(k) $ is not dependent on mass. this isn't obvious to me, since if it is of the form $ m k $ or something silly of this nature, then it will still drop out as $ k \rightarrow 0$.

I'm quite confused by two short paragraphs in Schwarz 28. He proves that $ Q = \int d^3 x J_0(x) = \int d^3 x \sum_m \frac{\partial L}{\partial \dot \phi_m} \frac{\delta \phi_m}{\delta \alpha} $ (where $ J_\mu $ is a conserved current) is a generator for the symmetry transformation, which is fine, and that $ Q | \Omega\rangle $ is degenerate with the ground state due to H commuting with Q due to charge conservation.

Fine, the issue is when he states $ | \pi(\bar p ) \rangle = \frac{-2i}{F} \int d^3 x exp(i \bar p \cdot \bar x ) J_0(x) | \Omega \rangle $. If I insert one of the definitions above into the below, we end up with something like $ \int d^3 x exp(i \bar p \cdot \bar x ) \sum_m \pi_m(x) \frac{d\phi_m}{d\alpha} $.

1: Is this just a definition, or is there a deeper mathematical reason? Intuitively, it makes sense since we expect $ p = 0 $ to produce $ | \pi(0 ) \rangle$ so it's just like we're moving it in momentum space.

2: Furthermore, he then states they have energy $ E(p) + E_0 $, which also confuses me: when applying H to the above, assuming we can pass it into the integral, I'd expect to see some kind of obvious separation like $ H( \int d^3 x J_0 (x) | \Omega\rangle ) + H (something else) $ to correspond to the form above, but this is also unclear to me on how to proceed.

3:I'm also confused on his final assertion, that since $ E(p) \rightarrow 0$ as $ p \rightarrow 0 $, (which makes sense), that there is a massless dispersion relation. this means that $E(k) $ is not dependent on mass. this isn't obvious to me, since if it is of the form $ m k $ or something silly of this nature, then it will still drop out as $ k \rightarrow 0$.

I'm quite confused by two short paragraphs in Schwarz 28. He proves that $$ Q = \int d^3 x J_0(x) = \int d^3 x \sum_m \frac{\partial L}{\partial \dot \phi_m} \frac{\delta \phi_m}{\delta \alpha} $$ (where $ J_\mu $ is a conserved current) is a generator for the symmetry transformation, which is fine, and that $ Q | \Omega\rangle $ is degenerate with the ground state due to $H$ commuting with $Q$ due to charge conservation.

Fine, the issue is when he states $$ | \pi(\bar p ) \rangle = \frac{-2i}{F} \int d^3 x exp(i \bar p \cdot \bar x ) J_0(x) | \Omega \rangle .$$ If I insert one of the definitions above into the below, we end up with something like $$ \int d^3 x exp(i \bar p \cdot \bar x ) \sum_m \pi_m(x) \frac{d\phi_m}{d\alpha} .$$

1. Is this just a definition, or is there a deeper mathematical reason? Intuitively, it makes sense since we expect $ p = 0 $ to produce $ | \pi(0 ) \rangle$ so it's just like we're moving it in momentum space.

2. Furthermore, he then states they have energy $ E(p) + E_0 $, which also confuses me: when applying $H$ to the above, assuming we can pass it into the integral, I'd expect to see some kind of obvious separation like $$ H( \int d^3 x J_0 (x) | \Omega\rangle ) + H (something else) $$ to correspond to the form above, but this is also unclear to me on how to proceed.

3. I'm also confused on his final assertion, that since $ E(p) \rightarrow 0$ as $ p \rightarrow 0 $, (which makes sense), that there is a massless dispersion relation. this means that $E(k) $ is not dependent on mass. this isn't obvious to me, since if it is of the form $ m k $ or something silly of this nature, then it will still drop out as $ k \rightarrow 0$.