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Is it possible to find the behavior of a field theory with a square well in the Lagrangian? Usually, we have polynomial terms, and one can argue (see e.g. in [physics SE q41065]) that the simplest case is $$ \mathcal{L}={\small \frac12}\partial_\mu\phi\,\partial^{\hspace{0.2pt}\mu}\!\phi - V(\phi) \quad \text{with} \quad V(\phi)={\small \frac12}\phi^2 $$ But while harmonic oscillator potentials are indeed often-used simple QM examples, the square-well potential is also quite popular. So what if we use a square well for $V$ (or equivalently $\phi^{\,p}$ in the $\phi^\infty$ limit?) $$ V(\phi) =\lim_{p\rightarrow\infty}\phi^{\,p} = \begin{cases} {\small 0 \ \ \ \ \text{if} \ \ -1<\phi<1 } \\ {\small \infty \ \ \text{otherwise} } \end{cases} $$ To find the ground state we could regularize on a lattice of $N$ points $x_n$ with field operators $\phi_n$ and construct the ground state wave functional $\Psi_0[\phi]$ as: (assuming a simple boson field) $$ \Psi_0[\phi] = \prod_n \cos{\small\frac{\pi}2}\phi_n $$ so simply the product of square-well ground states. Likewise, for a single-particle plane wave excitation with momentum $k$ one could try in the usual way: (assuming 1+1 dimensions) $$ \Psi_\text{1p}[\phi] = \sum_m e^{ik x_{\large m} } \Big(\sin \pi\phi_m \prod_{n\neq m} \cos{\small\frac{\pi}2}\phi_n\Big) $$ which has one first excited square-well state, distributed over the field points with the usual exponential phase factor. Unfortunately, however, it can easily be seen that this ansatz is no exact solution, since the whole problem is just the $N$-dimensional Schrodinger equation in an $N$-dimensional hypercube like this:

where the Hamiltonian, for lattice spacing $\varepsilon$, is: $$ H = \frac1N \sum_n \left[ -{\small\frac12}\frac{\partial^2}{\partial\phi_n^2} + {\small\frac12}{\frac{(\phi_n-\phi_{n-1})^2}{\varepsilon^2}} + V(\phi_n) \right] $$

Also the mechanism of creation and annihilation operators does not seem to be possible here (although the first excited square-well state $\sin \pi\phi$ can be related to the cosine-shaped ground state by multiplying with $\sin\frac{\pi}2\phi$, but that does not provide much help).

So it remains mysterious and I'm left wondering:

1. Can this field theory actually be consistently defined in the continuum limit (rescaling and renomalising as needed of course) or is there something that prevents this?
2. If it can be defined, what would be the properties? (Would the field acquire a mass? would there be a low-energy effective field theory with ordinary polynomial terms, perhaps $\phi^4$ theory?)

Is it possible to find the behavior of a field theory with a square well in the Lagrangian? Usually, we have polynomial terms, and one can argue (see e.g. in [physics SE q41065]) that the simplest case is $$ \mathcal{L}={\small \frac12}\partial_\mu\phi\,\partial^{\hspace{0.2pt}\mu}\!\phi - V(\phi) \quad \text{with} \quad V(\phi)={\small \frac12}\phi^2 $$ But while harmonic oscillator potentials are indeed often-used simple QM examples, the square-well potential is also quite popular. So what if we use a square well for $V$ (or equivalently $\phi^{\,p}$ in the $\phi^\infty$ limit?) $$ V(\phi) =\lim_{p\rightarrow\infty}\phi^{\,p} = \begin{cases} {\small 0 \ \ \ \ \text{if} \ \ -1<\phi<1 } \\ {\small \infty \ \ \text{otherwise} } \end{cases} $$ To find the ground state we could regularize on a lattice of $N$ points $x_n$ with field operators $\phi_n$ and try to construct the ground state wave functional $\Psi_0[\phi]$ as: (assuming a simple boson field) $$ \Psi_0[\phi] = \prod_n \cos{\small\frac{\pi}2}\phi_n $$ so simply the product of square-well ground states. Likewise, for a single-particle plane wave excitation with momentum $k$ one could try in the usual way: (assuming 1+1 dimensions) $$ \Psi_\text{1p}[\phi] = \sum_m e^{ik x_{\large m} } \Big(\sin \pi\phi_m \prod_{n\neq m} \cos{\small\frac{\pi}2}\phi_n\Big) $$ which has one first excited square-well state, distributed over the field points with the usual exponential phase factor. Unfortunately, however, it can easily be seen that this ansatz is no exact solution, since the whole problem is just the $N$-dimensional Schrodinger equation in an $N$-dimensional hypercube like this:

where the Hamiltonian, for lattice spacing $\varepsilon$, is: $$ H = \frac1N \sum_n \left[ -{\small\frac12}\frac{\partial^2}{\partial\phi_n^2} + {\small\frac12}{\frac{(\phi_n-\phi_{n-1})^2}{\varepsilon^2}} + V(\phi_n) \right] $$

Also the mechanism of creation and annihilation operators does not seem to be possible here (although the first excited square-well state $\sin \pi\phi$ can be related to the cosine-shaped ground state by multiplying with $\sin\frac{\pi}2\phi$, but that does not provide much help).

So it remains mysterious and I'm left wondering:

1. Can this field theory actually be consistently defined in the continuum limit (rescaling and renomalising as needed of course) or is there something that prevents this?
2. If it can be defined, what would be the properties? (Would the field acquire a mass? would there be a low-energy effective field theory with ordinary polynomial terms, perhaps $\phi^4$ theory?)

Is it possible to find the behavior of a field theory with a square well in the Lagrangian? Usually, we have polynomial terms, and one can argue (see e.g. in [physics SE q41065]) that the simplest case is $$\mathcal{L}={\small \frac12}\partial_\mu\phi\,\partial^\mu\!\phi - V(\phi) \quad \text{with} \quad V(\phi)={\small \frac12}\phi^2 $$ But while harmonic oscillator potentials are indeed often-used simple QM examples, the square-well potential is also quite popular. So what if we use a square well for $V$ (or equivalently $\phi^{\,p}$ in the $\phi^\infty$ limit?) $$ V(\phi) =\lim_{p\rightarrow\infty}\phi^{\,p} = \begin{cases} {\small 0 \ \ \ \ \text{if} \ \ -1<\phi<1 } \\ {\small \infty \ \ \text{otherwise} } \end{cases} $$ To find the ground state we could regularize on a lattice of $N$ points $x_n$ with field operators $\phi_n$ and construct the ground state wave functional $\Psi_0[\phi]$ as: (assuming a simple boson field) $$ \Psi_0[\phi] = \prod_n \cos{\small\frac{\pi}2}\phi_n $$ so simply the product of square-well ground states. Likewise, for a single-particle plane wave excitation with momentum $k$ one could try in the usual way: (assuming 1+1 dimensions) $$ \Psi_\text{1p}[\phi] = \sum_m e^{ik x_{\large m} } \Big(\sin \pi\phi_m \prod_{n\neq m} \cos{\small\frac{\pi}2}\phi_n\Big) $$ which has one first excited square-well state, distributed over the field points with the usual exponential phase factor. Unfortunately, however, it can easily be seen that this ansatz is no exact solution, since the whole problem is just the $N$-dimensional Schrodinger equation in an $N$-dimensional hypercube like this:

where the Hamiltonian, for lattice spacing $\varepsilon$, is: $$ H = \frac1N \sum_n \left[ -{\small\frac12}\frac{\partial^2}{\partial\phi_n^2} + {\small\frac12}{\frac{(\phi_n-\phi_{n-1})^2}{\varepsilon^2}} + V(\phi_n) \right] $$

Also the mechanism of creation and annihilation operators does not seem to be possible here (although the first excited square-well state $\sin \pi\phi$ can be related to the cosine-shaped ground state by multiplying with $\sin\frac{\pi}2\phi$, but that does not provide much help).

So it remains mysterious and I'm left wondering:

1. Can this field theory actually be consistently defined in the continuum limit (rescaling and renomalising as needed of course) or is there something that prevents this?
2. If it can be defined, what would be the properties? (Would the field acquire a mass? would there be a low-energy effective field theory with ordinary polynomial terms, perhaps $\phi^4$ theory?)

Is it possible to find the behavior of a field theory with a square well in the Lagrangian? Usually, we have polynomial terms, and one can argue (see e.g. in [physics SE q41065]) that the simplest case is $$ \mathcal{L}={\small \frac12}\partial_\mu\phi\,\partial^{\hspace{0.2pt}\mu}\!\phi - V(\phi) \quad \text{with} \quad V(\phi)={\small \frac12}\phi^2 $$ But while harmonic oscillator potentials are indeed often-used simple QM examples, the square-well potential is also quite popular. So what if we use a square well for $V$ (or equivalently $\phi^{\,p}$ in the $\phi^\infty$ limit?) $$ V(\phi) =\lim_{p\rightarrow\infty}\phi^{\,p} = \begin{cases} {\small 0 \ \ \ \ \text{if} \ \ -1<\phi<1 } \\ {\small \infty \ \ \text{otherwise} } \end{cases} $$ To find the ground state we could regularize on a lattice of $N$ points $x_n$ with field operators $\phi_n$ and construct the ground state wave functional $\Psi_0[\phi]$ as: (assuming a simple boson field) $$ \Psi_0[\phi] = \prod_n \cos{\small\frac{\pi}2}\phi_n $$ so simply the product of square-well ground states. Likewise, for a single-particle plane wave excitation with momentum $k$ one could try in the usual way: (assuming 1+1 dimensions) $$ \Psi_\text{1p}[\phi] = \sum_m e^{ik x_{\large m} } \Big(\sin \pi\phi_m \prod_{n\neq m} \cos{\small\frac{\pi}2}\phi_n\Big) $$ which has one first excited square-well state, distributed over the field points with the usual exponential phase factor. Unfortunately, however, it can easily be seen that this ansatz is no exact solution, since the whole problem is just the $N$-dimensional Schrodinger equation in an $N$-dimensional hypercube like this:

where the Hamiltonian, for lattice spacing $\varepsilon$, is: $$ H = \frac1N \sum_n \left[ -{\small\frac12}\frac{\partial^2}{\partial\phi_n^2} + {\small\frac12}{\frac{(\phi_n-\phi_{n-1})^2}{\varepsilon^2}} + V(\phi_n) \right] $$

Also the mechanism of creation and annihilation operators does not seem to be possible here (although the first excited square-well state $\sin \pi\phi$ can be related to the cosine-shaped ground state by multiplying with $\sin\frac{\pi}2\phi$, but that does not provide much help).

So it remains mysterious and I'm left wondering:

1. Can this field theory actually be consistently defined in the continuum limit (rescaling and renomalising as needed of course) or is there something that prevents this?
2. If it can be defined, what would be the properties? (Would the field acquire a mass? would there be a low-energy effective field theory with ordinary polynomial terms, perhaps $\phi^4$ theory?)

Is it possible to find the behavior of a field theory with a square well in the Lagrangian? Usually, we have polynomial terms, and one can argue (see e.g. in [physics SE q41065]) that the simplest case is $$\mathcal{L}={\small \frac12}\partial_\mu\phi\,\partial^\mu\!\phi - V(\phi) \quad \text{with} \quad V(\phi)={\small \frac12}\phi^2 $$ But while harmonic oscillator potentials are indeed often-used simple QM examples, the square-well potential is also quite popular. So what if we use a square well for $V$ (or equivalently $\phi^{\,p}$ in the $\phi^\infty$ limit?) $$ V(\phi) =\lim_{p\rightarrow\infty}\phi^{\,p} = \begin{cases} {\small 0 \ \ \ \ \text{if} \ \ -1<\phi<1 } \\ {\small \infty \ \ \text{otherwise} } \end{cases} $$ To find the ground state we could regularize on a lattice of $N$ points $x_n$ with field operators $\phi_n$ and construct the ground state wave functional $\Psi_0[\phi]$ as: (assuming a simple boson field) $$ \Psi_0[\phi] = \prod_n \cos\frac{\pi}{2}\phi_n $$ so simply the product of square-well ground states. Likewise, for a single-particle plane wave excitation with momentum $k$ one could try in the usual way: (assuming 1+1 dimensions) $$ \Psi_\text{1p}[\phi] = \sum_m e^{ik x_{\large m} } \left(\sin \pi\phi_m \prod_{n\neq m} \cos{\small\frac{\pi}2}\phi_n\right) $$ which has one first excited square-well state, distributed over the field points with the usual exponential phase factor. Unfortunately, however, it can easily be seen that this ansatz is no exact solution, since the whole problem is just the $N$-dimensional Schrodinger equation in an $N$-dimensional hypercube like this:

where the Hamiltonian, for lattice spacing $\varepsilon$, is: $$ H = \frac1N \sum_n \left[ -{\small\frac12}\frac{\partial^2}{\partial\phi_n^2} + {\small\frac12}{\frac{(\phi_n-\phi_{n-1})^2}{\varepsilon^2}} + V(\phi_n) \right] $$

Also the mechanism of creation and annihilation operators does not seem to be possible here (although the first excited square-well state $\sin \pi\phi$ can be related to the cosine-shaped ground state by multiplying with $\sin\frac{\pi}2\phi$, but that does not provide much help).

So it remains mysterious and I'm left wondering:

1. Can this field theory actually be consistently defined in the continuum limit (rescaling and renomalising as needed of course) or is there something that prevents this?
2. If it can be defined, what would be the properties? (Would the field acquire a mass? would there be a low-energy effective field theory with ordinary polynomial terms, perhaps $\phi^4$ theory?)

Is it possible to find the behavior of a field theory with a square well in the Lagrangian? Usually, we have polynomial terms, and one can argue (see e.g. in [physics SE q41065]) that the simplest case is $$\mathcal{L}={\small \frac12}\partial_\mu\phi\,\partial^\mu\!\phi - V(\phi) \quad \text{with} \quad V(\phi)={\small \frac12}\phi^2 $$ But while harmonic oscillator potentials are indeed often-used simple QM examples, the square-well potential is also quite popular. So what if we use a square well for $V$ (or equivalently $\phi^{\,p}$ in the $\phi^\infty$ limit?) $$ V(\phi) =\lim_{p\rightarrow\infty}\phi^{\,p} = \begin{cases} {\small 0 \ \ \ \ \text{if} \ \ -1<\phi<1 } \\ {\small \infty \ \ \text{otherwise} } \end{cases} $$ To find the ground state we could regularize on a lattice of $N$ points $x_n$ with field operators $\phi_n$ and construct the ground state wave functional $\Psi_0[\phi]$ as: (assuming a simple boson field) $$ \Psi_0[\phi] = \prod_n \cos{\small\frac{\pi}2}\phi_n $$ so simply the product of square-well ground states. Likewise, for a single-particle plane wave excitation with momentum $k$ one could try in the usual way: (assuming 1+1 dimensions) $$ \Psi_\text{1p}[\phi] = \sum_m e^{ik x_{\large m} } \Big(\sin \pi\phi_m \prod_{n\neq m} \cos{\small\frac{\pi}2}\phi_n\Big) $$ which has one first excited square-well state, distributed over the field points with the usual exponential phase factor. Unfortunately, however, it can easily be seen that this ansatz is no exact solution, since the whole problem is just the $N$-dimensional Schrodinger equation in an $N$-dimensional hypercube like this:

where the Hamiltonian, for lattice spacing $\varepsilon$, is: $$ H = \frac1N \sum_n \left[ -{\small\frac12}\frac{\partial^2}{\partial\phi_n^2} + {\small\frac12}{\frac{(\phi_n-\phi_{n-1})^2}{\varepsilon^2}} + V(\phi_n) \right] $$

Also the mechanism of creation and annihilation operators does not seem to be possible here (although the first excited square-well state $\sin \pi\phi$ can be related to the cosine-shaped ground state by multiplying with $\sin\frac{\pi}2\phi$, but that does not provide much help).

So it remains mysterious and I'm left wondering:

1. Can this field theory actually be consistently defined in the continuum limit (rescaling and renomalising as needed of course) or is there something that prevents this?
2. If it can be defined, what would be the properties? (Would the field acquire a mass? would there be a low-energy effective field theory with ordinary polynomial terms, perhaps $\phi^4$ theory?)