Lattice field theories and the continuum limit

This means that the different lattice theories are all in the same universality class, and their differences at finite lattice spacing are only finite-size effects. This universality class is characterized by the symmetries and interactions of the underlying continuum theory, and this is how the lattice theories are related. In summary, different lattice theories for QCD can lead to the same continuum theory after renormalization, and their differences at finite lattice spacing are only finite-size effects determined by the underlying continuum theory's symmetries and interactions.
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I have heard generally that it is possible to put different physical theories on a lattice and after renormalization get the same continuum theory. I mean, different lattice theories that lead to the same continuum theory. Is this true for, say, qcd, or other particle theories? Are there different varieties of lattice qcd that give the same continuum qcd theory? And if so, is it known how these lattice theories are related, besides their obvious shared macroscopic limit?

Thanks.
 
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Yes, there are many different "lattice actions" for QCD that all give the same continuum limit. In renormalization group language, the different actions differ by irrelevant terms (terms down by powers of the lattice spacing) that vanish as the lattice spacing goes to zero.
 

FAQ: Lattice field theories and the continuum limit

1. What is a lattice field theory?

A lattice field theory is a mathematical framework used to study the behavior of quantum fields on a discrete lattice. This approach is used in theoretical physics to simulate and understand the behavior of quantum systems, such as particle interactions and phase transitions.

2. How does a lattice field theory relate to the continuum limit?

In a lattice field theory, the lattice spacing is considered to be a finite value. As this spacing becomes smaller and approaches zero, the theory approaches the continuum limit, where the lattice becomes infinitely fine and the system can be described by continuous equations. The continuum limit is important because it allows for the comparison of lattice simulations with experimental results.

3. What are the advantages of using lattice field theories?

One of the main advantages of lattice field theories is that they allow for the simulation of complex quantum systems that cannot be solved analytically. This provides a powerful tool for studying the behavior of these systems and making predictions. Additionally, lattice field theories can be used to study systems at different energy scales, providing insights into the behavior of these systems at different scales.

4. How do lattice field theories differ from traditional quantum field theories?

Traditional quantum field theories are based on continuous space-time, while lattice field theories introduce a discretization of space-time. This means that in lattice field theories, space and time are represented by a finite number of points, rather than being continuous. This approach allows for the use of numerical methods to study systems that cannot be solved analytically.

5. What are some challenges in using lattice field theories?

One of the main challenges in lattice field theories is the computational complexity of simulating systems with a large number of particles. As the number of particles increases, the number of calculations required also increases, making simulations more time-consuming and resource-intensive. Another challenge is the need to extrapolate results from the discrete lattice to the continuum limit, which can introduce errors and uncertainties in the predictions made by the theory.

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