- #1
howl
- 16
- 1
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?
This guy did:Demystifier said: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?
Demystifier said:It seems that lattice path integrals are not used in condensed matter. Does someone know why is that?
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?
Is it a consequence of Euclidean metric? With Minkowski (+---) metric I don't think it's true, because thenking vitamin said:antihermiticity of [itex]\gamma^{\nu}D_{\nu}[/itex].
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.
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.
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.
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.
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.