# How to evaluate path integrals numerically?

• A
• howl
In summary: Yes, the antihermiticity is a consequence of the Euclidean metric. In Minkowski space, the relevant relation is \gamma^{\nu} D_{\nu}^{\dagger} = -\gamma^{0} D_{0}.

#### howl

Since we only know Gaussian integration, could one get Green's function numerically with interacting action. Usual perturbation theory is tedious and limited, could one get high accurate result with PC beyond perturbation?

https://en.m.wikipedia.org/wiki/Lattice_QCD

However, if you find perturbation theory tedious, lattice computations may not be your best go-to unless you really really like programming, running your code for weeks on a supercluster, and then resubmit because you found a small bug. I would not go near lattice computations, but I am very grateful that others do because they are essential for understanding QCD.

Demystifier and atyy
It seems that lattice path integrals are not used in condensed matter. Does someone know why is that?

Demystifier said:
It seems that lattice path integrals are not used in condensed matter. Does someone know why is that?
This guy did:
https://www.researchgate.net/publication/1943569_Lattice_path_integral_approach_to_the_one-dimensional_Kondo_model

Demystifier said:
It seems that lattice path integrals are not used in condensed matter. Does someone know why is that?

I'm not sure, but it may be because of the "sign problem": https://arxiv.org/abs/1105.1374.

Demystifier said:
It seems that lattice path integrals are not used in condensed matter. Does someone know why is that?

The path integral over a Euclidean lattice is used in quantum Monte Carlo simulations in condensed matter physics, but the sign problem can be a big issue.

atyy
Why does the sign problem not appear in QCD?

The sign problem appears in QCD at finite density, $\mu \neq 0$. Recall that the chemical potential is introduced by taking
$$H \rightarrow H - \mu Q$$
where $Q$ is the particle density (the U(1) Noether current). So for the Dirac equation, $Q = \int d^3x \, \bar{\psi} \gamma^0 \psi$, so the Euclidean Dirac action becomes
$$\mathcal{S} = -\int d^{4} x \bar{\psi} \left( \gamma^{\nu} D_{\nu} + \gamma^0 \mu + m \right) \psi$$
($D_{\nu}$ is the covariant derivative, so this analysis includes coupling to gauge fields). Then in the path integral,
$$Z = \int\mathcal{D}A \, \mathcal{D}\bar{\psi} \, \mathcal{D}\psi \, e^{-\mathcal{S}} = \int \mathcal{D}A \, \mathrm{det}\left( \gamma^{\nu} D_{\nu} - \gamma^0 \mu + m \right)$$

Now the issue is that the Dirac operator satisfies
$$\left( \gamma^{\nu} D_{\nu} + \gamma^0 \mu + m \right)^{\dagger} = \left( -\gamma^{\nu} D_{\nu} + \gamma^0 \mu + m \right) = \gamma^5 \left( \gamma^{\nu} D_{\nu} - \gamma^0 \mu + m \right) \gamma^5$$
so
$$\mathrm{det}\left( \gamma^{\nu} D_{\nu} - \gamma^0 \mu + m \right)^{\dagger} = \mathrm{det}\left( \gamma^{\nu} D_{\nu} + \gamma^0 \mu + m \right)$$
So the determinant for Dirac fermions is only real at $\mu = 0$. For many calculations in lattice QCD you work at zero density and this is ok, but it seems that QCD at finite density is a major subject of interest with very rich many-body physics at play. I don't know much about the field of finite-density QCD, but some searching found an interesting discussion in Section IV of https://arxiv.org/abs/1101.0109 which mentions experimental conditions where this physics should emerge.

Obviously, in context of condensed matter the above manipulations only hold for systems which are Dirac-like at low energies, but more generally one can relate the sign problem to systems whose Euclidean path integrals have non-positive-definite Boltzmann weights.

Demystifier and Orodruin
@king vitamin I have some technical questions about the last two lines of your derivation. What is the point of introducing ##\gamma^5## matrices and how do they dissappear in the last line? Does your last line imply that ##{\rm det}(-\gamma^{0\dagger})={\rm det}(\gamma^0)##? If so, how is that compatible with ##\gamma^{0\dagger}=\gamma^0##?

In the last line I am using $(\gamma^5)^{-1} = \gamma^5$ and the fact that determinants are invariant under similarity transforms,
$$\mathrm{det}(M) = \mathrm{det}(S^{-1}M S) \qquad \forall S \in \mathrm{GL}(n).$$
for $M$ an $n\times n$ matrix.

The point of doing this is to show what I intended to show, but with $\mu=0$ this constitutes a proof that the determinant of the Euclidean Dirac operator is real in spite of the antihermiticity of $\gamma^{\nu}D_{\nu}$.

Demystifier said:
Does your last line implies that det(−γ0†)=det(γ0)det(−γ0†)=det(γ0){\rm det}(-\gamma^{0\dagger})={\rm det}(\gamma^0)? If so, how is that compatible with γ0†=γ0γ0†=γ0\gamma^{0\dagger}=\gamma^0?

It does imply that, which is perfectly compatible with the relation you gave. Don't forget that determinants are not linear! $\mathrm{det}(aM) = a^n \mathrm{det}(M)$ for an $n\times n$ matrix.

Demystifier
king vitamin said:
antihermiticity of $\gamma^{\nu}D_{\nu}$.
Is it a consequence of Euclidean metric? With Minkowski (+---) metric I don't think it's true, because then
$$\gamma^{0\dagger}=\gamma^0 ,\;\; \gamma^{i\dagger}=-\gamma^i, \;\; D_{\nu}^{\dagger}=-D_{\nu}$$
so ##\gamma^{\nu}D_{\nu}## is neither hermitian nor anti-hermitian for Minkowski gamma matrices.

Indeed, the hermiticity properties of the Minkowski-signature gamma matrices are not even invariant under Lorentz transformations. Recall that boosts generators are not Hermitian, so the spinorial representation matrices $S$ corresponding to boosts are non-unitary, so even if $\gamma^{\mu}$ is (anti)hermitian, the matrix $S^{-1}\gamma^{\mu}S$ will not be in general. But in contrast to Spin(1,3), representations of the group Spin(4) can be chosen so that all elements are unitary so that (anti)hermiticity is preserved. (I believe you can choose all of the Euclidean gamma matrices to be antihermitian, and the arguments in my previous posts go through identically.) The fact that Spin(4) has this nice property which Spin(3,1) does not have is related to the compactness of the former.

Of course, Euclidean signature is important before even mentioning fermions because we need to get rid of that factor of $i$ sitting in front of the action in the path integral. Oscillating Boltzmann weights need to be avoided for computational efficiency. Of course, while Euclidean-time is ok if you just want static observables, you need to analytically continue if you want access to dynamics, and analytically continuing numerical results is ill-defined in general. So this is another major stumbling block for these methods.

Demystifier
Thanks, that was very illuminating!

## 1. How do I choose the appropriate numerical method for evaluating path integrals?

The choice of numerical method depends on the type of path integral and the complexity of the integrand. Some common methods include Monte Carlo integration, Gaussian quadrature, and Simpson's rule. It is important to carefully consider the properties of the integral and the desired level of accuracy before selecting a method.

## 2. What is the significance of discretizing the path in path integrals?

Discretizing the path involves dividing the integral into smaller segments and approximating the integral over each segment. This is necessary in numerical evaluation of path integrals as it allows for the use of numerical methods that are only applicable to finite intervals. The accuracy of the evaluation depends on the size of the segments chosen.

## 3. How do I handle singularities in the integrand when evaluating path integrals numerically?

Singularities in the integrand can lead to inaccuracies in the numerical evaluation of path integrals. One approach is to avoid the singularity by choosing a different path or by transforming the integral. Another approach is to use specialized numerical methods that can handle singularities, such as adaptive quadrature.

## 4. Can I use the same numerical method for all types of path integrals?

No, different types of path integrals require different numerical methods for accurate evaluation. For example, Monte Carlo integration is more suitable for high-dimensional integrals, while Gaussian quadrature may be more efficient for low-dimensional integrals. It is important to choose a method that is appropriate for the specific integral being evaluated.

## 5. How can I improve the accuracy of my numerical evaluation of path integrals?

To improve the accuracy, one can decrease the size of the discretized segments, use a more precise numerical method, or increase the number of samples in Monte Carlo integration. It is also important to check for errors in the implementation of the chosen method and to verify the results using other methods or analytical solutions if available.