Can Euclidean lattice quantum field theory be analytically continued into Minkowski space?

The paper argues that analytically continuing Euclidean lattice quantum field theory directly into Minkowski space via inverse Wick rotation is impossible due to nonlocal form factors introduced by discretization, necessitating a prior continuum limit.

The Gist

The Big Question: Can We Translate "Grid Physics" Back to "Real Physics"?

Imagine you are trying to understand how a complex machine works, but the machine is too fast and chaotic to study directly. So, you decide to build a slow-motion, black-and-white model of it. You take a photo of the machine every second, turn those photos into a grid (like a spreadsheet), and study the patterns in the grid. This is what physicists do with Euclidean Lattice Quantum Field Theory. They turn the smooth, continuous universe into a grid of points (a lattice) to make difficult calculations possible on supercomputers.

The big hope has always been: "If we solve the puzzle on this grid, can we just 'translate' the answer back into the real, smooth, colorful universe (Minkowski space) where we actually live?"

This paper says: No, you cannot simply translate it back.

The Core Problem: The "Pixelated" Universe

The authors argue that the moment you turn the smooth universe into a grid, you fundamentally break the rules that allow for that translation.

The Analogy of the Pixelated Photo:
Think of a smooth, high-definition video of a car driving down a road.

• Real World (Minkowski): The car moves smoothly. You can predict exactly where it will be at any fraction of a second.
• The Grid (Lattice): You chop the video into huge, blocky pixels. The car doesn't move smoothly anymore; it "jumps" from one pixel to the next.

In the real world, the car's movement is local. To know where the car is, you only need to know where it was a tiny fraction of a second ago.
In the grid world, because the car jumps from pixel to pixel, its movement becomes non-local. To understand the car's motion, you have to look at the "jump" it made. This jump acts like a "form factor" (a fancy math term for a rule that changes how things interact).

The Failed Translation: The "Wick Rotation"

Physicists have a magical tool called the Wick Rotation. Imagine the timeline of the universe is a piece of paper.

• In the "Grid World" (Euclidean), time is just another dimension, like width or height.
• In the "Real World" (Minkowski), time is different; it flows forward.

The Wick Rotation is like taking that piece of paper and rotating it 90 degrees to turn "width" back into "time." For smooth, continuous theories, this works perfectly.

The Paper's Discovery:
The authors show that if you try to rotate the grid paper, it tears.
Because the grid forces particles to "jump" between points, the math describing these jumps contains a hidden "explosive" factor.

• The Metaphor: Imagine trying to rotate a spinning top that is balanced on a needle. If the top is smooth, it spins fine. But if the top is made of jagged, jagged blocks (the grid), the moment you try to tilt it (rotate it), the jagged edges catch and the whole thing flies apart.

Mathematically, the "jaggedness" of the grid creates a factor that grows infinitely large when you try to rotate the time axis. The integral (the sum of all possibilities) blows up to infinity instead of settling down. Therefore, the translation is impossible.

The Only Way Out: Fix the Grid First

The paper concludes that you cannot do the rotation while the grid is still there.

• Wrong Order: Grid $\rightarrow$ Rotate $\rightarrow$ Real World. (This fails because the grid breaks the rotation).
• Right Order: Grid $\rightarrow$ Shrink the Grid to Zero (Continuum Limit) $\rightarrow$ Real World.

You must first make the pixels so small they disappear, restoring the smooth, local nature of the universe. Only then can you perform the rotation.

Why Does This Matter?

The authors point out two main consequences:

1. Mathematical Meaning: The "Feynman Path Integral" (the method used to calculate probabilities in quantum physics) is mathematically well-defined on the grid. But if you can't rotate it back to real time, we can't say for sure that this mathematical method actually describes our real, time-flowing universe. It might just be a useful trick for the grid, not a description of reality.
2. Lost Concepts: In the real world, we can talk about "semi-classical" processes (like a ball rolling down a hill). The paper suggests that because we can't translate the grid results back to real time, we lose the ability to identify these specific types of processes within the grid theory. The concept of "time flowing forward" in the way we experience it is lost in the grid.

Summary

The paper claims that Euclidean Lattice Quantum Field Theory is a dead end for direct translation. You cannot take the results from a computer simulation of a pixelated universe and simply "rotate" them to get the physics of our real, smooth universe. The act of pixelating the universe introduces a "jaggedness" that breaks the mathematical bridge (the Wick rotation) between the two worlds. To get real physics, you must first remove the pixels entirely.

Technical Summary

Technical Summary: Analytic Continuation of Euclidean Lattice QFT to Minkowski Space

Problem Statement
The paper addresses a fundamental issue in lattice gauge quantum field theory: whether the Euclidean lattice formulation can be directly analytically continued into Minkowski space ($R^{1,3}$) via the inverse Wick rotation. While lattice theory is the primary non-perturbative tool for studying strongly coupled systems (such as QCD), a widely held view suggests that the transition matrix allows one to relate the Euclidean lattice path integral to a quantum mechanical description in Minkowski space. The authors argue that this view overlooks a critical obstruction: the discretization of spacetime itself converts a local quantum field theory into a nonlocal theory with a form factor, rendering the standard Wick rotation infeasible without first taking the continuum limit ($a \to 0$).

Methodology
The authors employ a rigorous mathematical analysis of the discretization process, focusing on the simplest case of a non-interacting neutral scalar field in two-dimensional Euclidean spacetime ($E_2$) to avoid complications like fermion doubling. Their methodology involves:

1. Formalism of Discretization: They express the lattice action, where integration is replaced by summation and differentiation by finite differences, using continuous techniques. Specifically, they utilize the operator identity $\phi(\tau + a) = \exp(a \frac{d}{d\tau})\phi(\tau)$ to represent the shift between lattice sites.
2. Fourier Analysis: By transforming the discretized action into momentum space, they demonstrate that the interaction term acquires an additional factor, $\tilde{K}(p) \propto \exp(i\ell_\mu p^\mu)$, which acts as a nonlocal form factor. This identifies the lattice theory as a specific instance of a nonlocal form factor field theory (in the Meiman–Jaffe–Efimov classification).
3. Contour Integration Analysis: The authors attempt to perform the inverse Wick rotation ($\omega \to -ip_0$) by rotating the integration axis in the complex plane by $\pi/2$. They analyze the behavior of the integrand, specifically the term $|e^{i\omega(\tau+a)}|$, along arcs of infinite radius in the complex $\omega$ plane.
4. Convergence Testing: They evaluate whether the integrals over the connecting arcs in the first and fourth quadrants vanish, a necessary condition for the validity of the analytic continuation.

Key Contributions and Results
The paper establishes that the operations of taking the continuum limit and performing the Wick rotation are non-commutative. The specific findings are:

• Nonlocality of Lattice Theory: The discretization of spacetime introduces a form factor that makes the theory nonlocal. In the Fourier transform, this manifests as an exponential growth factor in the ultraviolet region.
• Failure of Wick Rotation: When analyzing the integral over an infinite-radius arc in the complex plane, the authors find that for certain conditions (specifically when $\tau < -|a|$), the integral diverges. Consequently, the integral over the arc connecting the Euclidean and pseudo-Euclidean domains does not vanish.
• Impossibility of Direct Continuation: Because the arc integrals do not vanish, the analytic continuation from $E_2$ to $R^{1,1}$ (and by extension $E_4$ to $R^{1,3}$) via inverse Wick rotation is mathematically impossible for a theory that retains the lattice spacing $a$.
• Continuum Limit Requirement: The authors conclude that one must first take the limit $a \to 0$ to restore locality and remove the nonlocal form factor before a valid analytic continuation to Minkowski space can be performed.

Significance and Claims
The paper claims two primary implications of this result:

1. Formal/Mathematical Significance: The Feynman functional integral measure is rigorously well-defined only in Euclidean spacetime. If direct analytic continuation were possible, it might allow for a mathematical definition of the measure in the original Minkowski formulation. The authors assert that since direct continuation is impossible, such a definition remains unfeasible.
2. Conceptual Significance: The inability to analytically continue from $E_4$ to $R^{1,3}$ implies that identifying specific processes in lattice theory that correspond to timelike momenta in intermediate states (characterized by tree diagrams and the semiclassical regime) is "hardly possible." Consequently, the very concept of a semiclassical regime is lost within the lattice framework unless the continuum limit is taken first.

The authors emphasize that this result clarifies why the "strategy of Constructive Quantum Field Theory" (finding the continuum limit in Euclidean space, verifying axioms, and then constructing the Minkowski theory) is necessary, particularly in four dimensions where exactly solvable models are absent. They explicitly state that their discussion indicates such analytical continuation is impossible without first taking the lattice theory to the continuum limit.