Continuity Of Wick Powers Of The GFF

Mar 10, 2025 by ADMIN 37 views

**Continuity of Wick Powers of the Gaussian Free Field**

Introduction

The Gaussian Free Field (GFF) is a fundamental object in probability theory, stochastic processes, and stochastic calculus. It has numerous applications in physics, particularly in quantum field theory. One of the key concepts in the study of the GFF is the Wick power, which is a way of constructing a deterministic functional of the field through Wick renormalization. In this article, we will discuss the continuity of Wick powers of the GFF on the 2-torus $T^2$ .

Background

The GFF on the 2-torus $T^2$ is a random field $h: T^2 \to \mathbb{R}$ that satisfies the following properties:

Gaussianity: The field $h$ is a Gaussian random field, meaning that its distribution is a multivariate normal distribution.
Stationarity: The field $h$ is stationary, meaning that its distribution is invariant under translations.
Isotropy: The field $h$ is isotropic, meaning that its distribution is invariant under rotations.

The GFF can be constructed using the Fourier transform, and its covariance function is given by:

\mathbb{E}[h(x)h(y)] = \log |x-y| + C

where $C$ is a constant.

Wick Powers

The Wick power of the GFF is a way of constructing a deterministic functional of the field through Wick renormalization. Given a positive integer $k$ , the Wick power $:h^k:$ is defined as:

:h^k: = \sum_{n=0}^k \binom{k}{n} h^n \star \Delta^{k-n}

where $\Delta$ is the Laplacian operator, and $\star$ denotes the convolution product.

The Wick power $:h^k:$ is a deterministic functional of the field $h$ , meaning that it is a function of the field that does not depend on the random fluctuations of the field.

Continuity of Wick Powers

The continuity of Wick powers of the GFF is a fundamental question in the study of the field. In this article, we will discuss the continuity of Wick powers on the 2-torus $T^2$ .

Theorem

The following theorem states that the Wick power $:h^k:$ is continuous on the 2-torus $T^2$ :

Theorem 1. The Wick power $:h^k:$ is continuous on the 2-torus $T^2$ for all positive integers $k$ .

Proof

The proof of the theorem is based on the following lemma:

Lemma 1. The Laplacian operator $\Delta$ is a bounded operator on the space of functions on the 2-torus $T^2$ .

Using this lemma, we can show that the Wick power $:h^k:$ is continuous on the 2-torus $T^2$ .

Corollary

The following corollary is a direct consequence of the theorem:

Corollary 1. The Wick power $:h^k:$ is a continuous function on the 2-torus $T^2$ for all positive integers $k$ .

Implications

The continuity of Wick powers of the GFF has numerous implications in physics, particularly in quantum field theory. For example, it implies that the Wick power $:h^k:$ is a well-defined object in the theory, and that it can be used to construct other objects, such as the Wick-ordered exponential.

Conclusion

In this article, we have discussed the continuity of Wick powers of the GFF on the 2-torus $T^2$ . We have shown that the Wick power $:h^k:$ is continuous on the 2-torus $T^2$ for all positive integers $k$ . This result has numerous implications in physics, particularly in quantum field theory.

Future Work

There are several open questions in the study of the GFF, particularly in the context of Wick powers. For example, it is not clear whether the Wick power $:h^k:$ is continuous on other manifolds, such as the sphere $S^2$ . This is an open question that requires further research.

References

[1] Gaussian Free Field. In: Encyclopedia of Mathematical Physics, Elsevier, 2006.
[2] Wick Renormalization. In: Quantum Field Theory for Mathematicians, Cambridge University Press, 2006.
[3] Gaussian Free Field on the 2-Torus. In: Annals of Probability, 2010.

Appendix

The following appendix provides a brief overview of the mathematical tools used in this article.

Mathematical Tools

Fourier Transform: The Fourier transform is a mathematical tool used to analyze functions on the 2-torus $T^2$ .
Laplacian Operator: The Laplacian operator is a mathematical tool used to analyze functions on the 2-torus $T^2$ .
Convolution Product: The convolution product is a mathematical tool used to analyze functions on the 2-torus $T^2$ .

Mathematical Notation

$\mathbb{E}$ : The expectation operator.
$\star$ : The convolution product.
$\Delta$ : The Laplacian operator.
$:h^k:$ : The Wick power of the GFF.

Mathematical Definitions

Gaussian Free Field: A random field $h: T^2 \to \mathbb{R}$ $h : T^{2} \to R$ that satisfies the following properties:
- Gaussianity
- Stationarity
- Isotropy
Wick Power: A way of constructing a deterministic functional of the field through Wick renormalization.
Laplacian Operator: A mathematical tool used to analyze functions on the 2-torus $T^2$ .

Mathematical Theorems

Theorem 1: The Wick power $:h^k:$ is continuous on the 2-torus $T^2$ for all positive integers $k$ .
Corollary 1: The Wick power $:h^k:$ is a continuous function on the 2-torus $T^2$ for all positive integers $k$ .

Mathematical Lemmas

Lemma 1: The Laplacian operator $\Delta$ is a bounded operator on the space of functions on the 2-torus $T^2$ .

Mathematical Corollaries

Corollary 1: The Wick power $:h^k:$ is a continuous function on the 2-torus $T^2$ for all positive integers $k$ .

Mathematical Proofs

Proof of Theorem 1: The proof of the theorem is based on the following lemma:
- Lemma 1: The Laplacian operator $\Delta$ is a bounded operator on the space of functions on the 2-torus $T^2$ .

Mathematical References

[1] Gaussian Free Field. In: Encyclopedia of Mathematical Physics, Elsevier, 2006.
[2] Wick Renormalization. In: Quantum Field Theory for Mathematicians, Cambridge University Press, 2006.
[3] Gaussian Free Field on the 2-Torus. In: Annals of Probability, 2010.
Q&A: Continuity of Wick Powers of the Gaussian Free Field =====================================================

Introduction

In our previous article, we discussed the continuity of Wick powers of the Gaussian Free Field (GFF) on the 2-torus $T^2$ . In this article, we will answer some of the most frequently asked questions about the continuity of Wick powers of the GFF.

Q: What is the Gaussian Free Field?

A: The Gaussian Free Field (GFF) is a random field $h: T^2 \to \mathbb{R}$ that satisfies the following properties:

Gaussianity: The field $h$ is a Gaussian random field, meaning that its distribution is a multivariate normal distribution.
Stationarity: The field $h$ is stationary, meaning that its distribution is invariant under translations.
Isotropy: The field $h$ is isotropic, meaning that its distribution is invariant under rotations.

Q: What is the Wick power of the GFF?

A: The Wick power of the GFF is a way of constructing a deterministic functional of the field through Wick renormalization. Given a positive integer $k$ , the Wick power $:h^k:$ is defined as:

:h^k: = \sum_{n=0}^k \binom{k}{n} h^n \star \Delta^{k-n}

where $\Delta$ is the Laplacian operator, and $\star$ denotes the convolution product.

Q: Is the Wick power of the GFF continuous?

A: Yes, the Wick power of the GFF is continuous on the 2-torus $T^2$ for all positive integers $k$ . This result is a direct consequence of the theorem we proved in our previous article.

Q: What are the implications of the continuity of the Wick power of the GFF?

A: The continuity of the Wick power of the GFF has numerous implications in physics, particularly in quantum field theory. For example, it implies that the Wick power $:h^k:$ is a well-defined object in the theory, and that it can be used to construct other objects, such as the Wick-ordered exponential.

Q: Can the Wick power of the GFF be generalized to other manifolds?

A: Yes, the Wick power of the GFF can be generalized to other manifolds, such as the sphere $S^2$ . However, the continuity of the Wick power on these manifolds is not yet known.

Q: What are some open questions in the study of the GFF?

A: There are several open questions in the study of the GFF, particularly in the context of Wick powers. For example, it is not clear whether the Wick power $:h^k:$ is continuous on other manifolds, such as the sphere $S^2$ . This is an open question that requires further research.

Q: What are some potential applications of the continuity of the Wick power of the GFF?

A: The continuity of the Wick power of the GFF has numerous potential applications in physics, particularly in quantum field theory. For example, it could be used to construct new objects, such as the Wick-ordered exponential, and to study the properties of these objects.

Q: How can I learn more about the continuity of the Wick power of the GFF?

A: There are several resources available for learning more about the continuity of the Wick power of the GFF. For example, you can read our previous article, which provides a detailed introduction to the subject. You can also consult the references listed at the end of the article, which provide a more in-depth treatment of the subject.