As I understand it, a relativistic quantum field is an operator-valued function of spacetime that transforms under some finite dimensional irreducible representation of the Lorentz* group: \begin{equation} U(\Lambda)\Psi^{\alpha}U(\Lambda^{-1}) = D(\Lambda^{-1})^\alpha{}_{\beta}\Psi^{\beta}(\Lambda x) \end{equation} Particle states (which I will write as $|\mathbf{p},\lambda \rangle = a^\dagger (\mathbf{p},\lambda) |0\rangle$, with $\mathbf{p}$ being the 3-momentum and $\lambda$ being the helicity), on the other hand, must preserve unitarity due to Quantum Mechanics and therefore should transform under unitary representations of the Poincaré* group

\begin{equation} \Lambda|\mathbf{p},\lambda\rangle = |\mathbf{p}',\lambda'\rangle D^{s}(R(\Lambda,p))^{\lambda'}{}_{\lambda} \end{equation}

where $p'^{\mu} = \Lambda^{\mu}{}_{\nu}p^\nu$ and $\lambda $s are the helicities. Note that $\alpha, \beta$ are the Lorentz indices. The reason fields are introduced is because we can embed internal particle degrees of freedom transforming under an irreducible unitary representation (and hence infinite-dimensional) in a finite dimensional representation as

\begin{equation} \Psi^{\alpha}(x) = \sum_{\lambda}\int \tilde{d}p[a(\mathbf{p},\lambda)u^{\alpha}(\mathbf{p},\lambda)e^{ipx} + \text{ negative energy term}] \end{equation} where $\tilde{d}p$ is the Lorentz invariant integration measure. My questions are

1. How does one make contact with quantum mechanics? The book I am reading (Wu Ki Tung Group Theory in Physics, Chapter 10) says that the wavefunction for a state $|\phi\rangle$ is given by \begin{equation} \phi^\alpha(x) = \langle 0 | \Psi^{\alpha}(x) |\phi\rangle. \end{equation} So I assume this object describes the particle in the position basis. According the Peskin and Schroeder (chapter 2, eq. 2.42), the field operator creates a particle at position $x$ as such. But then what is the wavefunctional of the system, i.e., the object that describes the probability density corresponding to a particular configuration of the fields all throughout spacetime.

2. This is probably very naive but the book talks about relating the Lorentz symmetry of the fields to the Poincaré symmetry of the one-particle states. But I thought that fields are required to transform under the full Poincaré group, since Matthew D. Schwartz in his QFT and the Standard Model says that (chapter 8 titled Spin 1 and Gauge Invariance, page 110) scalar fields at all points in spacetime form a representation of translations since $\phi(x) \rightarrow \phi(x+a)$ under the action of translations. So I would expect that a natural demand to make is that the fields transform under finite dimensional representations of the Poincaré group.

As I understand it, a relativistic quantum field is an operator-valued function of spacetime that transforms under some finite dimensional irreducible representation of the Lorentz* group: \begin{equation} U(\Lambda)\Psi^{\alpha}U(\Lambda^{-1}) = D(\Lambda^{-1})^\alpha{}_{\beta}\Psi^{\beta}(\Lambda x) \end{equation} Particle states (which I will write as $|\mathbf{p},\lambda \rangle = a^\dagger (\mathbf{p},\lambda) |0\rangle$, with $\mathbf{p}$ being the 3-momentum and $\lambda$ being the helicity), on the other hand, must preserve unitarity due to Quantum Mechanics and therefore should transform under unitary representations of the Poincaré* group

\begin{equation} \Lambda|\mathbf{p},\lambda\rangle = |\mathbf{p}',\lambda'\rangle D^{s}(R(\Lambda,p))^{\lambda'}{}_{\lambda} \end{equation}

where $p'^{\mu} = \Lambda^{\mu}{}_{\nu}p^\nu$ and $\lambda $s are the helicities. Note that $\alpha, \beta$ are the Lorentz indices. The reason fields are introduced is because we can embed internal particle degrees of freedom transforming under an irreducible unitary representation (and hence infinite-dimensional) in a finite dimensional representation as

\begin{equation} \Psi^{\alpha}(x) = \sum_{\lambda}\int \tilde{d}p[a(\mathbf{p},\lambda)u^{\alpha}(\mathbf{p},\lambda)e^{ipx} + \text{ negative energy term}] \end{equation} where $\tilde{d}p$ is the Lorentz invariant integration measure. My question is:

How does one make contact with quantum mechanics? The book I am reading (Wu Ki Tung Group Theory in Physics, Chapter 10) says that the wavefunction for a state $|\phi\rangle$ is given by \begin{equation} \phi^\alpha(x) = \langle 0 | \Psi^{\alpha}(x) |\phi\rangle. \end{equation} So I assume this object describes the particle in the position basis. According the Peskin and Schroeder (chapter 2, eq. 2.42), the field operator creates a particle at position $x$ as such. But then what is the wavefunctional of the system, i.e., the object that describes the probability density corresponding to a particular configuration of the fields all throughout spacetime.

As I understand it, a relativistic quantum field is an operator-valued function of spacetime that transforms under some finite dimensional irreducible representation of the Lorentz* group: \begin{equation} U(\Lambda)\Psi^{\alpha}U(\Lambda^{-1}) = D(\Lambda^{-1})^\alpha{}_{\beta}\Psi^{\beta}(\Lambda x) \end{equation} Particle states (which I will write as $|\mathbf{p},\lambda \rangle = a^\dagger (\mathbf{p},\lambda) |0\rangle$, with $\mathbf{p}$ being the 3-momentum and $\lambda$ being the helicity), on the other hand, must preserve unitarity due to Quantum Mechanics and therefore should transform under unitary representations of the Poincare* group

\begin{equation} \Lambda|\mathbf{p},\lambda\rangle = |\mathbf{p}',\lambda'\rangle D^{s}(R(\Lambda,p))^{\lambda'}{}_{\lambda} \end{equation}

where $p'^{\mu} = \Lambda^{\mu}{}_{\nu}p^\nu$ and $\lambda $s are the helicities. Note that $\alpha, \beta$ are the Lorentz indices. The reason fields are introduced is because we can embed internal particle degrees of freedom transforming under an irreducible unitary representation (and hence infinite-dimensional) in a finite dimensional representation as

\begin{equation} \Psi^{\alpha}(x) = \sum_{\lambda}\int \tilde{d}p[a(\mathbf{p},\lambda)u^{\alpha}(\mathbf{p},\lambda)e^{ipx} + \text{ negative energy term}] \end{equation} where $\tilde{d}p$ is the Lorentz invariant integration measure. My questions are

1. How does one make contact with quantum mechanics? The book I am reading (Wu Ki Tung Group Theory in Physics, Chapter 10) says that the wavefunction for a state $|\phi\rangle$ is given by \begin{equation} \phi^\alpha(x) = \langle 0 | \Psi^{\alpha}(x) |\phi\rangle. \end{equation} So I assume this object describes the particle in the position basis. According the Peskin and Schroeder (chapter 2, eq. 2.42), the field operator creates a particle at position $x$ as such. But then what is the wavefunctional of the system, i.e., the object that describes the probability density corresponding to a particular configuration of the fields all throughout spacetime.

2. This is probably very naive but the book talks about relating the Lorentz symmetry of the fields to the Poincare symmetry of the one-particle states. But I thought that fields are required to transform under the full Poincare group, since Matthew D. Schwartz in his QFT and the Standard Model says that (chapter 8 titled Spin 1 and Gauge Invariance, page 110) scalar fields at all points in spacetime form a representation of translations since $\phi(x) \rightarrow \phi(x+a)$ under the action of translations. So I would expect that a natural demand to make is that the fields transform under finite dimensional representations of the Poincare group.

As I understand it, a relativistic quantum field is an operator-valued function of spacetime that transforms under some finite dimensional irreducible representation of the Lorentz* group: \begin{equation} U(\Lambda)\Psi^{\alpha}U(\Lambda^{-1}) = D(\Lambda^{-1})^\alpha{}_{\beta}\Psi^{\beta}(\Lambda x) \end{equation} Particle states (which I will write as $|\mathbf{p},\lambda \rangle = a^\dagger (\mathbf{p},\lambda) |0\rangle$, with $\mathbf{p}$ being the 3-momentum and $\lambda$ being the helicity), on the other hand, must preserve unitarity due to Quantum Mechanics and therefore should transform under unitary representations of the Poincaré* group

\begin{equation} \Lambda|\mathbf{p},\lambda\rangle = |\mathbf{p}',\lambda'\rangle D^{s}(R(\Lambda,p))^{\lambda'}{}_{\lambda} \end{equation}

where $p'^{\mu} = \Lambda^{\mu}{}_{\nu}p^\nu$ and $\lambda $s are the helicities. Note that $\alpha, \beta$ are the Lorentz indices. The reason fields are introduced is because we can embed internal particle degrees of freedom transforming under an irreducible unitary representation (and hence infinite-dimensional) in a finite dimensional representation as

\begin{equation} \Psi^{\alpha}(x) = \sum_{\lambda}\int \tilde{d}p[a(\mathbf{p},\lambda)u^{\alpha}(\mathbf{p},\lambda)e^{ipx} + \text{ negative energy term}] \end{equation} where $\tilde{d}p$ is the Lorentz invariant integration measure. My questions are

1. How does one make contact with quantum mechanics? The book I am reading (Wu Ki Tung Group Theory in Physics, Chapter 10) says that the wavefunction for a state $|\phi\rangle$ is given by \begin{equation} \phi^\alpha(x) = \langle 0 | \Psi^{\alpha}(x) |\phi\rangle. \end{equation} So I assume this object describes the particle in the position basis. According the Peskin and Schroeder (chapter 2, eq. 2.42), the field operator creates a particle at position $x$ as such. But then what is the wavefunctional of the system, i.e., the object that describes the probability density corresponding to a particular configuration of the fields all throughout spacetime.

2. This is probably very naive but the book talks about relating the Lorentz symmetry of the fields to the Poincaré symmetry of the one-particle states. But I thought that fields are required to transform under the full Poincaré group, since Matthew D. Schwartz in his QFT and the Standard Model says that (chapter 8 titled Spin 1 and Gauge Invariance, page 110) scalar fields at all points in spacetime form a representation of translations since $\phi(x) \rightarrow \phi(x+a)$ under the action of translations. So I would expect that a natural demand to make is that the fields transform under finite dimensional representations of the Poincaré group.

As I understand it, a relativistic quantum field is an operator-valued function of spacetime that transforms under some finite dimensional irreducible representation of the Lorentz* group: \begin{equation} U(\Lambda)\Psi^{\alpha}U(\Lambda^{-1}) = D(\Lambda^{-1})^\alpha{}_{\beta}\Psi^{\beta}(\Lambda x) \end{equation} Particle states (which I will write as $|\mathbf{p},\lambda \rangle = a^\dagger (\mathbf{p},\lambda) |0\rangle$, with $\mathbf{p}$ being the 3-momentum and $\lambda$ being the helicity), on the other hand, must preserve unitarity due to Quantum Mechanics and therefore should transform under unitary representations of the Poincare* group

\begin{equation} \Lambda|\mathbf{p},\lambda\rangle = |\mathbf{p}',\lambda'\rangle D^{s}(R(\Lambda,p))^{\lambda'}{}_{\lambda} \end{equation}

where $p'^{\mu} = \Lambda^{\mu}{}_{\nu}p^\nu$ and $\lambda $s are the helicities. Note that $\alpha, \beta$ are the Lorentz indices. The reason fields are introduced is because we can embed internal particle degrees of freedom transforming under an irreducible unitary representation (and hence infinite-dimensional) in a finite dimensional representation as

\begin{equation} \Psi^{\alpha}(x) = \sum_{\lambda}\int \tilde{d}p[a(\mathbf{p},\lambda)u^{\alpha}(\mathbf{p},\lambda)e^{ipx} + \text{ negative energy term}] \end{equation} where $\tilde{d}p$ is the Lorentz invariant integration measure. My questions are

1. How does one make contact with quantum mechanics. The book I am reading (Wu Ki Tung Group Theory in Physics, Chapter 10) says that the wavefunction for a state $|\phi\rangle$ is given by \begin{equation} \phi^\alpha(x) = \langle 0 | \Psi^{\alpha}(x) |\phi\rangle. \end{equation} So I assume this object describes the particle in the position basis. According the Peskin and Schroeder (chapter 2, eq. 2.42), the field operator creates a particle at position $x$ as such. But then what is the wavefunctional of the system, i.e., the object that describes the probability density corresponding to a particular configuration of the fields all throughout spacetime.

2. This is probably very naive but the book talks about relating the Lorentz symmetry of the fields to the Poincare symmetry of the one-particle states. But I thought that fields are required to transform under the full Poincare group, since Matthew D. Schwartz in his QFT and the Standard Model says that (chapter 8 titled Spin 1 and Gauge Invariance, page 110) scalar fields at all points in spacetime form a representation of translations since $\phi(x) \rightarrow \phi(x+a)$ under the action of translations. So I would expect that a natural demand to make is that the fields transform under finite dimensional representations of the Poincare group.

As I understand it, a relativistic quantum field is an operator-valued function of spacetime that transforms under some finite dimensional irreducible representation of the Lorentz* group: \begin{equation} U(\Lambda)\Psi^{\alpha}U(\Lambda^{-1}) = D(\Lambda^{-1})^\alpha{}_{\beta}\Psi^{\beta}(\Lambda x) \end{equation} Particle states (which I will write as $|\mathbf{p},\lambda \rangle = a^\dagger (\mathbf{p},\lambda) |0\rangle$, with $\mathbf{p}$ being the 3-momentum and $\lambda$ being the helicity), on the other hand, must preserve unitarity due to Quantum Mechanics and therefore should transform under unitary representations of the Poincare* group

\begin{equation} \Lambda|\mathbf{p},\lambda\rangle = |\mathbf{p}',\lambda'\rangle D^{s}(R(\Lambda,p))^{\lambda'}{}_{\lambda} \end{equation}

where $p'^{\mu} = \Lambda^{\mu}{}_{\nu}p^\nu$ and $\lambda $s are the helicities. Note that $\alpha, \beta$ are the Lorentz indices. The reason fields are introduced is because we can embed internal particle degrees of freedom transforming under an irreducible unitary representation (and hence infinite-dimensional) in a finite dimensional representation as

\begin{equation} \Psi^{\alpha}(x) = \sum_{\lambda}\int \tilde{d}p[a(\mathbf{p},\lambda)u^{\alpha}(\mathbf{p},\lambda)e^{ipx} + \text{ negative energy term}] \end{equation} where $\tilde{d}p$ is the Lorentz invariant integration measure. My questions are

1. How does one make contact with quantum mechanics? The book I am reading (Wu Ki Tung Group Theory in Physics, Chapter 10) says that the wavefunction for a state $|\phi\rangle$ is given by \begin{equation} \phi^\alpha(x) = \langle 0 | \Psi^{\alpha}(x) |\phi\rangle. \end{equation} So I assume this object describes the particle in the position basis. According the Peskin and Schroeder (chapter 2, eq. 2.42), the field operator creates a particle at position $x$ as such. But then what is the wavefunctional of the system, i.e., the object that describes the probability density corresponding to a particular configuration of the fields all throughout spacetime.

2. This is probably very naive but the book talks about relating the Lorentz symmetry of the fields to the Poincare symmetry of the one-particle states. But I thought that fields are required to transform under the full Poincare group, since Matthew D. Schwartz in his QFT and the Standard Model says that (chapter 8 titled Spin 1 and Gauge Invariance, page 110) scalar fields at all points in spacetime form a representation of translations since $\phi(x) \rightarrow \phi(x+a)$ under the action of translations. So I would expect that a natural demand to make is that the fields transform under finite dimensional representations of the Poincare group.