Discretizing Scalar Field Theory on a Euclidean Lattice

1. From Quantum Dynamics to Statistical Mechanics

The key insight that makes lattice simulations possible is a deep connection between quantum field theory and statistical mechanics. This connection is made explicit by performing a Wick rotation, where we analytically continue from real time, $t$, to imaginary time, $\tau$, by setting $t=-i\tau$.

Let's see the consequence of this. In quantum mechanics, the time-evolution operator is $e^{-iHt/\hslash }$. After a Wick rotation, this becomes $e^{-H\tau /\hslash }$. If we identify the imaginary time extent $\tau$ with the inverse temperature $\beta =1/T$, this factor becomes $e^{-\beta H}$, which you will recognize as the Boltzmann weight from statistical mechanics!

This transformation has a profound effect:

• The oscillatory phase factor $e^{iS/\hslash }$ in the quantum mechanical path integral, which is numerically challenging, becomes a real, positive-definite probability weight $e^{-S_E/\hslash }$, where $S_E$ is the Euclidean action.
• The Minkowski metric ${\eta }_{\mu \nu }=\text{diag}(1,-1,-1,-1)$ becomes the Euclidean metric ${\delta }_{\mu \nu }=\text{diag}(1,1,1,1)$. Spacetime becomes a 4-dimensional Euclidean space.
• Quantum mechanical expectation values of operators can be calculated as thermal averages in a corresponding statistical mechanics system.

To see how this formal correspondence is established, please study the following resources.

The crucial takeaway is that by moving to Euclidean spacetime, we have transformed a quantum field theory problem into a classical statistical mechanics problem in four dimensions. The task of calculating quantum observables is now equivalent to calculating ensemble averages for a statistical system whose probability distribution is given by $e^{-S_E}$.

2. Discretizing Spacetime and Fields

Our goal is to evaluate the path integral numerically. However, a computer cannot handle continuous functions or infinite spacetime. The next step is to discretize the theory, replacing the continuum with a finite grid, or lattice.

This discretization involves three simple replacements:

1. Spacetime: The continuous coordinate $x^{\mu }$ is replaced by a set of discrete points on a hypercubic lattice: $x^{\mu }\to n^{\mu }a$, where $n^{\mu }$ is a vector of integers and $a$ is the lattice spacing.
2. Fields: The field $\varphi (x)$, which is a function over continuous spacetime, becomes a set of values $\varphi (n)$ defined only at the lattice sites $n$.
3. Derivatives and Integrals:
   • The integral over spacetime is replaced by a sum over all lattice sites: $\int d^{4}x\to a^4{\sum }_n$.
   • Derivatives are replaced by finite differences. The simplest choice for a first derivative is the forward difference: ${\partial }_{\mu }\varphi (x)\to \frac{\varphi (n+\hat{\mu })-\varphi (n)}{a}$, where $\hat{\mu }$ is a unit vector in the $\mu$ direction.

3. The Discretized Scalar Field Action

Now we can apply these rules to formulate the action for a real scalar field theory (specifically, ${\varphi }^4$ theory) on the lattice. The continuum Euclidean action is:

Here, $({\partial }_{\mu }\varphi {)}^2={\sum }_{\mu =0}^3({\partial }_{\mu }\varphi )({\partial }_{\mu }\varphi )$ since the metric is Euclidean.

Let's discretize this term by term:

• Potential Terms: The mass and interaction terms are "local" (they only depend on the field at a single point $x$). Their discretization is straightforward: $$\int d^{4}x\left(\frac{1}{2}{m}_0^2\varphi (x{)}^2+\frac{{\lambda }_0}{4!}\varphi (x{)}^4\right)⟶a^4\sum _n\left(\frac{1}{2}{m}_0^2\varphi (n{)}^2+\frac{{\lambda }_0}{4!}\varphi (n{)}^4\right)$$
• Kinetic Term: This term involves derivatives and couples neighboring sites. Using the symmetric second derivative you just worked out, we can relate it to the continuum version via an "integration by parts" on the lattice (a summation by parts). The continuum kinetic term $\int ({\partial }_{\mu }\varphi {)}^2$ can be written as $-\int \varphi ({\partial }_{\mu }^2\varphi )$ (ignoring boundary terms). The discretized version of the operator $-{\partial }_{\mu }^2$ is precisely what you derived.

Let's see the final form in the resources.

From the resources, a standard form for the discretized ${\varphi }^4$ action can be written as:

Or, using a common re-parameterization found in simulation literature (as in resource LINK), it can be written in terms of dimensionless fields $\phi$ and couplings $\kappa$ and $\lambda$:

While the forms look different, they describe the same physics and are related by a redefinition of the field and couplings. The crucial part is that the action is now a function of a finite number of variables—the field values $\varphi (n)$ at each lattice site. The path integral becomes a high-dimensional, but well-defined, integral:

This is the central result for today. We have transformed the abstract, operator-based formulation of a quantum field theory into a concrete numerical problem.

Conclusion

We have taken a major step into the world of non-perturbative QFT. By combining the Wick rotation with a discretization of spacetime, we have reformulated a quantum field theory as a classical statistical mechanics problem on a 4D lattice.

Key Takeaways:

• Wick Rotation ($t\to -i\tau$) transforms the quantum evolution operator $e^{-iHt}$ into a statistical Boltzmann weight $e^{-\beta H}$, allowing for a probabilistic interpretation.
• This turns the Minkowski path integral with its complex weight $e^{iS}$ into a Euclidean path integral with a real, positive weight $e^{-S_E}$.
• Discretization replaces continuous spacetime with a lattice, continuous fields with values at lattice sites, derivatives with finite differences, and integrals with sums.
• The Euclidean action for a scalar field on the lattice becomes a well-defined function of the field values at all sites, coupling nearest neighbors through the kinetic term.
• The problem of calculating physical observables in QFT is now equivalent to computing expectation values in a statistical system using the path integral $Z=\int [d\varphi ]e^{-S_L[\varphi ]}$.

Preview of the Next Lesson:

We now have our theory on a lattice, and the path integral has become a (very!) high-dimensional integral. The natural question is: how do we actually compute it? Direct integration is impossible for any realistically sized lattice. In our next lesson, we will answer this by learning about the workhorse algorithm of lattice field theory: the Metropolis-Hastings algorithm. We will learn how to use this Monte Carlo method to generate field configurations distributed according to the Boltzmann weight $e^{-S_L}$ and use them to calculate physical observables like correlation functions.