New Classes of Solutions for Euclidean Scalar Field Theories
Abstract
:1. Introduction
- (1)
- is continuously differentiable;
- (2)
- ;
- (3)
- is not positive definite;
- (4)
- , where and are positive and .
2. The Role of Scale Invariance
3. The Role of Mass
4. Solutions of the Massless Equation
5. Classes of Instanton Solutions
5.1. Solutions with Nonpolynomial
5.2. Solutions with Polynomial
5.3. Conformally Invariant Equations
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
1 | It was shown in [27] that massive solutions exist in our case for . |
2 | No new solutions are obtained by generalizing to Laurent polynomials. If F is a polynomial of degree p, the term , which is of order , needs to be for F to satisfy (18). |
3 | A conformal field theory has an infinite number of conservation laws at [33]. |
4 | One should be aware of the 1-2-3-infinity fallacy in which one makes a general conclusion based on a few examples. A nice example of small-n behavior giving an incorrect answer for a general n is provided by the integral , where . The integrals for , but this relation fails at . However, we have carefully checked that there are no such problems with the solutions presented in this paper. |
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Bender, C.M.; Sarkar, S. New Classes of Solutions for Euclidean Scalar Field Theories. Universe 2024, 10, 72. https://doi.org/10.3390/universe10020072
Bender CM, Sarkar S. New Classes of Solutions for Euclidean Scalar Field Theories. Universe. 2024; 10(2):72. https://doi.org/10.3390/universe10020072
Chicago/Turabian StyleBender, Carl M., and Sarben Sarkar. 2024. "New Classes of Solutions for Euclidean Scalar Field Theories" Universe 10, no. 2: 72. https://doi.org/10.3390/universe10020072