Undecidable problems in quantum field theory

Abstract: We point out that some questions in quantum field theory are undecidable in a precise mathematical sense. More concretely, it will be demonstrated that there is no algorithm answering whether a given 2d supersymmetric Lagrangian theory breaks supersymmetry or not. It will also be shown that there is a specific 2d supersymmetric Lagrangian theory which breaks supersymmetry if and only if the standard Zermelo-Fraenkel set theory with the axiom of choice is consistent, which can never be proved or disproved as the consequence of G\"odel's second incompleteness theorem. The article includes a brief and informal introduction to the phenomenon of undecidability and its previous appearances in theoretical physics.

• The paper shows that determining supersymmetry breaking in 2D Wess-Zumino models is undecidable, being equivalent to solving a Diophantine equation.
• It establishes a reduction from key quantum field theory problems to the Turing machine halting problem, revealing inherent computational limits.
• The study bridges undecidability in classical mechanics, quantum spin systems, and supersymmetric theories, highlighting fundamental algorithmic constraints.

This paper, "Undecidable problems in quantum field theory" (2203.16689), points out that some fundamental questions in theoretical physics, specifically within quantum field theory, are undecidable in a precise mathematical sense. Undecidability in this context means there is no general algorithm that can answer the question for every possible input system within a given class. The paper distinguishes between a "generic" undecidability (no single algorithm for all cases) and a "specific" undecidability (a concrete system exists for which the property cannot be proved or disproved within a standard mathematical framework like ZFC set theory, as a consequence of Gödel's incompleteness theorems).

The core mechanism for demonstrating undecidability in physics systems, as described in the paper, is to show that the problem can be reduced to a known undecidable problem from computer science or mathematical logic. The most fundamental example is the Halting Problem for Turing machines: given a program, will it halt (finish) or run forever? There is no algorithm that can determine this for all possible programs. Another related undecidable problem is the solvability of Diophantine equations (polynomial equations with integer coefficients where integer solutions are sought), known as Hilbert's 10th problem, which was proven undecidable by Matiyasevich (building on work by Davis, Putnam, and Robinson). The solvability of a Diophantine equation can be constructed to be equivalent to whether a specific Turing machine halts. Gödel's incompleteness theorems show that in any consistent formal system strong enough to describe arithmetic, there are statements that can neither be proved nor disproved within that system, and the consistency of the system itself is one such statement. This implies the existence of a specific Turing machine whose halting behavior is formally undecidable within standard mathematics.

The paper reviews previous instances where undecidability has been found in theoretical physics:

1. Classical Mechanical Systems: Early works [GerochHartle, Moore, daCostaDoria] showed that predicting specific outcomes of classical systems or determining if certain functions within their description are zero can be undecidable. This was achieved by embedding the logic of a Turing machine into the dynamics of the classical system.
2. Quantum Spin Systems: More recent work [2dNature, 2dLong] demonstrated that properties of the ground state of 2D quantum spin systems with finite-range, translationally invariant interactions can be undecidable. Specifically, the question of whether such a system is gapless in the infinite volume limit was shown to be undecidable. The implementation involves constructing a Hamiltonian $H_\xi$ for a spin system corresponding to a Turing machine $\xi$, such that the ground state energy behavior depends on whether $\xi$ halts. By combining this Hamiltonian with auxiliary Hamiltonians, properties like gaplessness or the specific symmetry-protected topological phase of the ground state can be made equivalent to the halting problem. The construction requires significant ingenuity to ensure the resulting Hamiltonian adheres to the constraints of local, translationally invariant interactions on a lattice.

The paper's main novel contribution is demonstrating undecidability in 2D $\mathcal{N}=(2,2)$ supersymmetric Lagrangian theories, specifically Wess-Zumino models. The property shown to be undecidable is whether such a theory spontaneously breaks supersymmetry (i.e., whether it has supersymmetric vacua).

The implementation relies on the undecidability of Diophantine equations. For any given Turing machine $\xi$, a Diophantine equation $P_\xi(x_1, \ldots, x_k) = 0$ can be constructed such that it has integer solutions if and only if $\xi$ halts. The paper then considers a 2D $\mathcal{N}=(2,2)$ Wess-Zumino model with chiral superfields $Y$, $Z_1, \ldots, Z_k$, and $X_1, \ldots, X_k$. A superpotential $W_\xi$ is constructed based on the Diophantine equation $P_\xi$: $$W_\xi = Y P_\xi(X_1,\ldots,X_k)^2 + \sum_a Z_a (\sin 2\pi i X_a)^2$$.

In a Wess-Zumino model, supersymmetric vacua exist if and only if there are field configurations for which the F-term conditions, given by $\partial W/\partial \phi = 0$ for all fields $\phi$, are simultaneously satisfied. For the constructed superpotential $W_\xi$:

• $$\partial W_\xi / \partial Z_a = (\sin 2\pi i X_a)^2$$. Setting this to zero requires $\sin 2\pi i X_a = 0$, which implies $X_a$ must be an integer for all $a=1, \ldots, k$.
• $$\partial W_\xi / \partial Y = P_\xi(X_1,\ldots,X_k)^2$$. Setting this to zero requires $P_\xi(X_1,\ldots,X_k) = 0$.

Thus, the F-term conditions $\partial W_\xi/\partial Z_a=0$ and $\partial W_\xi/\partial Y=0$ are satisfied if and only if the Diophantine equation $P_\xi(X_1, \ldots, X_k)=0$ has a solution where $X_1, \ldots, X_k$ are integers. If these conditions are met, the $\partial W_\xi/\partial X_a=0$ conditions are automatically satisfied (as $W_\xi$ depends on $X_a$ only through $P_\xi^2$ and $\sin^2$, which are zero when $X_a \in \mathbb{Z}$ and $P_\xi=0$), without imposing further constraints on $Y$ or $Z_a$.

Therefore, this Wess-Zumino model has supersymmetric vacua if and only if the Diophantine equation $P_\xi(X_1, \ldots, X_k)=0$ has a solution in integers. Consequently, the theory breaks supersymmetry (i.e., has no supersymmetric vacuum) if and only if the Diophantine equation $P_\xi$ has no integer solution.

Since there is no algorithm to decide whether an arbitrary Diophantine equation has integer solutions (which is equivalent to the Halting Problem for the associated Turing machine $\xi$), there is no algorithm to decide whether an arbitrary 2D $\mathcal{N}=(2,2)$ supersymmetric Wess-Zumino model (given by its superpotential $W_\xi$) breaks supersymmetry.

Furthermore, by taking the Diophantine equation $P_{\xi_{\mathcal{ZFC}}}=0$ corresponding to a Turing machine $\xi_{\mathcal{ZFC}}$ whose halting is undecidable within $\mathcal{ZFC}$ set theory (such as one that halts if and only if $\mathcal{ZFC}$ is inconsistent), one can construct a specific Wess-Zumino model whose supersymmetry breaking is undecidable within $\mathcal{ZFC}$.

The paper notes that encoding Diophantine equations into the structure of Wess-Zumino models is relatively straightforward because the superpotential can be an arbitrary holomorphic function. This contrasts with potentially more constrained classes of theories, such as 4D renormalizable quantum field theories, where interactions are restricted to low-degree polynomials. The practical implication for theoretical physicists is the understanding that certain fundamental questions about physical systems may not be solvable by any finite algorithmic procedure, regardless of future theoretical or computational advancements.