Essentially nothing like the what you describe in the more detailed form of your question exists for the important reason that your "intuitive conception" of a hypothetical non-linear state space is incorrect.

The linear structure in a quantum theory HAS NOTHING TO DO with whether or not the equations of motion are linear. It is an exact feature that persists independent of any interactions. Indeed the Hilbert space arises from the fact that a configuration of a quantum system is given as a complex valued linear functional from the vector space $V$ which has a basis in bijective correspondence with the possible classical states of the system. (Typically these functionals are required to be square normalizable in a suitable sense but this issue is secondary to the present discussion.) Linear functionals of course always have the structure of a vector space and as a result so too does the state space of a quantum mechanical theory including in particular quantum field theory.

As a specific example consider quantum mechanics of a single particle moving on a line $\mathbb{R}$. The space of classical configurations is of course the real line, meaning exactly that to specify the all classical information means to specify a function $x(t)$ which tells you at which point $x$ the particle can be found at time $t$. Quantum mechanically this is modified as follows, we introduce a vector space $V$ with a basis of states in bijective correspondence with the set of all allowed classical configurations. Thus in this example $V$ has a basis for each point in $\mathbb{R}$. The Hilbert space of the theory is then the dual space (over $\mathbb{C}$) to $V$. This is true no matter what non-linear interactions may be occurring, and a wide variety of exactly solvable non-linear models exist. In this example the dual space is spanned by Dirac delta functions, typically denoted by $|x\rangle$ which indicates the functional that takes a non-zero value on the basis element corresponding to point $x$ and zero at all other basis elements associated to points $y\neq x$. As a result the Hilbert space can be described as (suitably normalizable) distributions from $\mathbb{R}$ to $\mathbb{C}$. The interpretation of an element of this Hilbert space $\Psi(x): \mathbb{R}\longrightarrow \mathbb{C}$ is that the square of its norm specifies the probability distribution for the quantum mechanical particle to be observed at the position $x$. The time evolution of the wave function $\Psi(x)$ is then always governed by the linear Schrodinger equation.

In a quantum field theory the abstract structure is the same, however $V$ is typically a much larger vector space since it has a basis in correspondence with fields, i.e. functions from space to say $\mathbb{R}$.

Essentially nothing like the what you describe in the more detailed form of your question exists for the important reason that your "intuitive conception" of a hypothetical non-linear state space is incorrect.

The linear structure in a quantum theory HAS NOTHING TO DO with whether or not the equations of motion are linear. It is an exact feature that persists independent of any interactions. Indeed the Hilbert space arises from the fact that a configuration of a quantum system is given as a complex valued linear functional from the vector space $V$ which has a basis in bijective correspondence with the possible classical states of the system. (Typically these functionals are required to be square normalizable in a suitable sense but this issue is secondary to the present discussion.) Linear functionals of course always have the structure of a vector space and as a result so too does the state space of a quantum mechanical theory including in particular quantum field theory.

As a specific example consider quantum mechanics of a single particle moving on a line $\mathbb{R}$. The space of classical configurations is of course the real line, meaning exactly that to specify the all classical information means to specify a function $x(t)$ which tells you at which point $x$ the particle can be found at time $t$. Quantum mechanically this is modified as follows. We introduce a vector space $V$ with a basis of states in bijective correspondence with the set of all allowed classical configurations. Thus in this example $V$ has a basis for each point in $\mathbb{R}$. The Hilbert space of the theory is then the dual space (over $\mathbb{C}$) to $V$. This is true no matter what non-linear interactions may be occurring, and a wide variety of exactly solvable non-linear models exist. In this example the dual space is spanned by Dirac delta functions, typically denoted by $|x\rangle$ which indicates the functional that takes a non-zero value on the basis element corresponding to point $x$ and zero at all other basis elements associated to points $y\neq x$. As a result the Hilbert space can be described as (suitably normalizable) distributions from $\mathbb{R}$ to $\mathbb{C}$. The interpretation of an element of this Hilbert space $\Psi(x): \mathbb{R}\longrightarrow \mathbb{C}$ is that the square of its norm specifies the probability distribution for the quantum mechanical particle to be observed at the position $x$. The time evolution of the wave function $\Psi(x)$ is then always governed by the linear Schrodinger equation.

In a quantum field theory the abstract structure is the same, however $V$ is typically a much larger vector space since it has a basis in correspondence with fields, i.e. functions from space to say $\mathbb{R}$.

Essentially nothing like the what you describe in the more detailed form of your question exists for the important reason that your "intuitive conception" of a hypothetical non-linear state space is incorrect.

The linear structure in a quantum theory HAS NOTHING TO DO with whether or not the equations of motion are linear. It is an exact feature that persists independent of any interactions. Indeed the Hilbert space arises from the fact that a configuration of a quantum system is given as a complex valued linear functional from the vector space $V$ which has a basis in bijective correspondence with the possible classical states of the system. (Typically these functionals are required to be square normalizable in a suitable sense but this issue is secondary to the present discussion.) Linear functionals of course always have the structure of a vector space and as a result so too does the state space of a quantum mechanical theory including in particular quantum field theory.

As a specific example consider quantum mechanics of a single particle moving on a line $\mathbb{R}$. The space of classical configurations is of course the real line, meaning exactly that to specify the all classical information means to specify a function $x(t)$ which tells you at which point $x$ the particle can be found at time $t$. Quantum mechanically this is modified as follows, we introduce a vector space $V$ with a basis of states in bijective correspondence with the set of all allowed classical configurations. Of course in this example $V$ is just $\mathbb{R}$. The Hilbert space of the theory is then the space of square normalizable complex valued functions on $V$. This is true no matter what non-linear interactions may be occurring, and a wide variety of exactly solvable non-linear models exist. The interpretation of an element of this Hilbert space $\Psi(x): \mathbb{R}\longrightarrow \mathbb{C}$ is that the square of its norm specifies the probability distribution for the quantum mechanical particle to be observed at the position $x$. The time evolution of the wave function $\Psi(x)$ is then always governed by the linear Schrodinger equation.

In a quantum field theory the abstract structure is the same, however $V$ is typically infinite dimensional since it has a basis in correspondence with fields, i.e. functions from space to say $\mathbb{R}$.

Essentially nothing like the what you describe in the more detailed form of your question exists for the important reason that your "intuitive conception" of a hypothetical non-linear state space is incorrect.

The linear structure in a quantum theory HAS NOTHING TO DO with whether or not the equations of motion are linear. It is an exact feature that persists independent of any interactions. Indeed the Hilbert space arises from the fact that a configuration of a quantum system is given as a complex valued linear functional from the vector space $V$ which has a basis in bijective correspondence with the possible classical states of the system. (Typically these functionals are required to be square normalizable in a suitable sense but this issue is secondary to the present discussion.) Linear functionals of course always have the structure of a vector space and as a result so too does the state space of a quantum mechanical theory including in particular quantum field theory.

As a specific example consider quantum mechanics of a single particle moving on a line $\mathbb{R}$. The space of classical configurations is of course the real line, meaning exactly that to specify the all classical information means to specify a function $x(t)$ which tells you at which point $x$ the particle can be found at time $t$. Quantum mechanically this is modified as follows, we introduce a vector space $V$ with a basis of states in bijective correspondence with the set of all allowed classical configurations. Thus in this example $V$ has a basis for each point in $\mathbb{R}$. The Hilbert space of the theory is then the dual space (over $\mathbb{C}$) to $V$. This is true no matter what non-linear interactions may be occurring, and a wide variety of exactly solvable non-linear models exist. In this example the dual space is spanned by Dirac delta functions, typically denoted by $|x\rangle$ which indicates the functional that takes a non-zero value on the basis element corresponding to point $x$ and zero at all other basis elements associated to points $y\neq x$. As a result the Hilbert space can be described as (suitably normalizable) distributions from $\mathbb{R}$ to $\mathbb{C}$. The interpretation of an element of this Hilbert space $\Psi(x): \mathbb{R}\longrightarrow \mathbb{C}$ is that the square of its norm specifies the probability distribution for the quantum mechanical particle to be observed at the position $x$. The time evolution of the wave function $\Psi(x)$ is then always governed by the linear Schrodinger equation.

In a quantum field theory the abstract structure is the same, however $V$ is typically a much larger vector space since it has a basis in correspondence with fields, i.e. functions from space to say $\mathbb{R}$.