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Equations updated to MO2.0.

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edited Jul 6, 2013 at 17:42

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Your intuition is not entirely without basis, though. Spelling it out, also shows how the standard formalism of QFT (Wightman or any related one) already accommodates non-linear interactions. One can define the following hierarchy of $n$-point functions (sometimes called Wightman functions): \begin{align} W^0_\Psi &= \langle 0|\Psi\rangle \\ W^1_\Psi(t_1,x_1) &= \langle 0|\hat{\phi}(t_1,x_1)|\Psi\rangle \\ W^2_\Psi(t_1,x_1;t_2,x_2) &= \langle 0|\hat{\phi}(t_1,x_1)\hat{\phi}(t_2,x_2)|\Psi\rangle \\ & \cdots \end{align}\begin{align} W^0_\Psi &= \langle 0|\Psi\rangle \\ W^1_\Psi(t_1,x_1) &= \langle 0|\hat{\phi}(t_1,x_1)|\Psi\rangle \\ W^2_\Psi(t_1,x_1;t_2,x_2) &= \langle 0|\hat{\phi}(t_1,x_1)\hat{\phi}(t_2,x_2)|\Psi\rangle \\ & \cdots \end{align} It is a fundamental result in QFT (known under different names, such as the Wightman reconstruction theorem, multiparticle representation of states, or simply second quantization) that knowledge of all the $W^n_\Psi$ is completely equivalent to the knowledge of $|\Psi\rangle$.

These Wightman functions, by virtue of the Heisenberg equations of motion, satisfy the following infinite dimensional hierarchical system of equations \begin{align} \square_{t,x} W^1_\Psi(t,x) &= \lambda W^3_\Psi(t,x;t,x;t,x) + \text{(n-ord)} \\ \square_{t,x} W^2_\Psi(t,x;t_1,x_1) &= \lambda W^4_\Psi(t,x;t,x;t,x;t_1,x_1) + \text{(n-ord)} \\ \square_{t,x} W^2_\Psi(t_1,x_1;t,x) &= \lambda W^4_\Psi(t_1,x_1;t,x;t,x;t,x) + \text{(n-ord)} \\ & \cdots \end{align}\begin{align} \square_{t,x} W^1_\Psi(t,x) &= \lambda W^3_\Psi(t,x;t,x;t,x) + \text{(n-ord)} \\ \square_{t,x} W^2_\Psi(t,x;t_1,x_1) &= \lambda W^4_\Psi(t,x;t,x;t,x;t_1,x_1) + \text{(n-ord)} \\ \square_{t,x} W^2_\Psi(t_1,x_1;t,x) &= \lambda W^4_\Psi(t_1,x_1;t,x;t,x;t,x) + \text{(n-ord)} \\ & \cdots \end{align} I'm being a bit sloppy with coincidence limits here. The Wightman functions are singular if any two spacetime points in their arguments coincide, the terms labeled (n-ord) represent the necessary regulating subtractions to make this limit finite. This necessary regularization also explains why the non-linear terms in the equations of motion needed normal ordering.

Your intuition is not entirely without basis, though. Spelling it out, also shows how the standard formalism of QFT (Wightman or any related one) already accommodates non-linear interactions. One can define the following hierarchy of $n$-point functions (sometimes called Wightman functions): \begin{align} W^0_\Psi &= \langle 0|\Psi\rangle \\ W^1_\Psi(t_1,x_1) &= \langle 0|\hat{\phi}(t_1,x_1)|\Psi\rangle \\ W^2_\Psi(t_1,x_1;t_2,x_2) &= \langle 0|\hat{\phi}(t_1,x_1)\hat{\phi}(t_2,x_2)|\Psi\rangle \\ & \cdots \end{align} It is a fundamental result in QFT (known under different names, such as the Wightman reconstruction theorem, multiparticle representation of states, or simply second quantization) that knowledge of all the $W^n_\Psi$ is completely equivalent to the knowledge of $|\Psi\rangle$.

These Wightman functions, by virtue of the Heisenberg equations of motion, satisfy the following infinite dimensional hierarchical system of equations \begin{align} \square_{t,x} W^1_\Psi(t,x) &= \lambda W^3_\Psi(t,x;t,x;t,x) + \text{(n-ord)} \\ \square_{t,x} W^2_\Psi(t,x;t_1,x_1) &= \lambda W^4_\Psi(t,x;t,x;t,x;t_1,x_1) + \text{(n-ord)} \\ \square_{t,x} W^2_\Psi(t_1,x_1;t,x) &= \lambda W^4_\Psi(t_1,x_1;t,x;t,x;t,x) + \text{(n-ord)} \\ & \cdots \end{align} I'm being a bit sloppy with coincidence limits here. The Wightman functions are singular if any two spacetime points in their arguments coincide, the terms labeled (n-ord) represent the necessary regulating subtractions to make this limit finite. This necessary regularization also explains why the non-linear terms in the equations of motion needed normal ordering.

Your intuition is not entirely without basis, though. Spelling it out, also shows how the standard formalism of QFT (Wightman or any related one) already accommodates non-linear interactions. One can define the following hierarchy of $n$-point functions (sometimes called Wightman functions): \begin{align} W^0_\Psi &= \langle 0|\Psi\rangle \\ W^1_\Psi(t_1,x_1) &= \langle 0|\hat{\phi}(t_1,x_1)|\Psi\rangle \\ W^2_\Psi(t_1,x_1;t_2,x_2) &= \langle 0|\hat{\phi}(t_1,x_1)\hat{\phi}(t_2,x_2)|\Psi\rangle \\ & \cdots \end{align} It is a fundamental result in QFT (known under different names, such as the Wightman reconstruction theorem, multiparticle representation of states, or simply second quantization) that knowledge of all the $W^n_\Psi$ is completely equivalent to the knowledge of $|\Psi\rangle$.

These Wightman functions, by virtue of the Heisenberg equations of motion, satisfy the following infinite dimensional hierarchical system of equations \begin{align} \square_{t,x} W^1_\Psi(t,x) &= \lambda W^3_\Psi(t,x;t,x;t,x) + \text{(n-ord)} \\ \square_{t,x} W^2_\Psi(t,x;t_1,x_1) &= \lambda W^4_\Psi(t,x;t,x;t,x;t_1,x_1) + \text{(n-ord)} \\ \square_{t,x} W^2_\Psi(t_1,x_1;t,x) &= \lambda W^4_\Psi(t_1,x_1;t,x;t,x;t,x) + \text{(n-ord)} \\ & \cdots \end{align} I'm being a bit sloppy with coincidence limits here. The Wightman functions are singular if any two spacetime points in their arguments coincide, the terms labeled (n-ord) represent the necessary regulating subtractions to make this limit finite. This necessary regularization also explains why the non-linear terms in the equations of motion needed normal ordering.

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created Apr 15, 2012 at 15:22

Igor Khavkine

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If I read your updated question correctly, you are asking whether people have considered non-linear modifications of quantum mechanics in order to accommodate interacting QFTs. I'm sure someone, somewhere has, but that's certainly not mainstream thought in QFT research, either on the mathematics or theoretical physics sides. Consider the analogous question in the quantum mechanics of particles: do non-linear equations of motion require a non-linear modification of quantum mechanics? The answer is most certainly No.

Without going into generalities, the Hydrogen atom and the double-well potential are prominent examples of systems with non-linear (Heisenberg) equations of motion that live perfectly well within the standard quantum formalism (states form a linear Hilbert space, observables are linear operators on states, time evolution is unitary on states in the Schroedinger picture and conjugation by unitary operators in the Heisenberg picture). When going from particle mechanics to field theory, what changes is the number of space-time dimensions, not the type of non-linearities in the equations of motion. So there is no mathematical reason to expect a non-linear modification of quantum mechanics in the transition.

Now, a few words about your intuition regarding states as solutions to the equations of motion. Unfortunately, it is somewhat off the mark. As you should be aware, relativistic QFT is usually discussed in the Heisenberg picture. This means that it is the field operators $\hat{\phi}(t,x)$ that obey the possibly non-linear equations of motion. For example, $\square\hat{\phi}(t,x) - \lambda{:}\hat{\phi}^3(t,x){:}=0$, where $\square$ is the wave operator and the colons denote normal ordering. On the other hand, states are just elements $|\Psi\rangle$ of an abstract Hilbert space (with the vacuum state $|0\rangle$ singled out by Poincaré invariance), entirely independent of spacetime coordinates. At this point, it should be clear why states have nothing to do with the equations of motion.

Your intuition is not entirely without basis, though. Spelling it out, also shows how the standard formalism of QFT (Wightman or any related one) already accommodates non-linear interactions. One can define the following hierarchy of $n$-point functions (sometimes called Wightman functions): \begin{align} W^0_\Psi &= \langle 0|\Psi\rangle \\ W^1_\Psi(t_1,x_1) &= \langle 0|\hat{\phi}(t_1,x_1)|\Psi\rangle \\ W^2_\Psi(t_1,x_1;t_2,x_2) &= \langle 0|\hat{\phi}(t_1,x_1)\hat{\phi}(t_2,x_2)|\Psi\rangle \\ & \cdots \end{align} It is a fundamental result in QFT (known under different names, such as the Wightman reconstruction theorem, multiparticle representation of states, or simply second quantization) that knowledge of all the $W^n_\Psi$ is completely equivalent to the knowledge of $|\Psi\rangle$.

These Wightman functions, by virtue of the Heisenberg equations of motion, satisfy the following infinite dimensional hierarchical system of equations \begin{align} \square_{t,x} W^1_\Psi(t,x) &= \lambda W^3_\Psi(t,x;t,x;t,x) + \text{(n-ord)} \\ \square_{t,x} W^2_\Psi(t,x;t_1,x_1) &= \lambda W^4_\Psi(t,x;t,x;t,x;t_1,x_1) + \text{(n-ord)} \\ \square_{t,x} W^2_\Psi(t_1,x_1;t,x) &= \lambda W^4_\Psi(t_1,x_1;t,x;t,x;t,x) + \text{(n-ord)} \\ & \cdots \end{align} I'm being a bit sloppy with coincidence limits here. The Wightman functions are singular if any two spacetime points in their arguments coincide, the terms labeled (n-ord) represent the necessary regulating subtractions to make this limit finite. This necessary regularization also explains why the non-linear terms in the equations of motion needed normal ordering.

If $\lambda=0$, the theory is non-interacting, then each of the above equations for the $W^n_\Psi$ becomes self-contained (independent of $n$-point functions of different order) and identical to the now linear equations of motion. At this point it should be clear how your intuition does in fact apply to the states of a non-interacting QFT. States $|\Psi\rangle$ can be put into correspondence with multiparticle "wave functions" solving the linear equations of motion (which are actually the Wightman functions $W^n_\Psi$).

Finally, when it comes to trying to construct models of QFT, people usually just concentrate on the Wightman functions associated to the vacuum state, $W^n_0 = \langle 0|\cdots|0\rangle$, which are sufficient to reconstruct the corresponding $n$-point functions for all other states. In short, the standard approaches to constructive QFT already incorporate non-linear interactions in a natural way. And non-linear modifications to the quantum mechanical formalism are simply a whole different, independent topic.