What are the Anomalies in Schwarz-Green and Yang-Mills Equations?

[Orion1]

String Theory: (TOE)

Schwarz-Green Anomaly Cancellation Equasions (1984):

$$S_o = \int d^o xe \left[- \left( \frac{1}{2 K^2} \right) R - \left( \frac{1}{K^2} \right) \left( \varphi^-2 \right) \vartheta_r \varphi \vartheta^n \varphi - \left( \frac{1}{4 g^2} \right) \right]...$$

$$... \left[ \left( \varphi^-3 \right) F_\mu ^o F^\mu - \left( \frac{ 3 K^2}{2 g^4} \right) \left( \varphi^-2 \right) H_p H^.ky \right]$$

$$H = dB + W_y ^o - W_l ^o$$
$$S_a = dA + \left[ A_1 \Lambda \right]$$
$$S_w = d \Theta + \left[ W_1 \Theta \right]$$
$$S_b = -tr \left( A_1 d \Lambda \right) + tr \left( W d \Theta \right)$$

Gravitational Anomaly:
$$\left( \frac{n496}{64} \right) \left[ \frac{1}{5870} trR^6 + \frac{ 1}{4379} trR^2 trR^4 + \frac{ 1}{10346} \left( trR^2 \right) ^3 \right]...$$
$$... + \frac{ 1}{864} trR^2 trR^4 + \frac{ 1}{1536} \left( trR^2 \right) ^3$$

Yang-Mills Anomaly:
$$- \frac{ 1}{15} \left( p - 32 \right) trF^6 +15 \left( p - 2 \right) tr F^2 \left( \left( p - 8 \right) trF^4 + 3 \left(trF^2 \right) ^2 \right)$$

$$p = 32$$
$$n = \frac{ 1}{2} p \left( p - 1 \right) \left( for SO \left( p \right) \right)$$
$$n = \frac{1}{2} \left( 32 \right) \left( 31 \right)$$
$$n = 496$$

Note! These equations have not been compiled completely accurately with the actual equations. If anyone has more accurate equations, please post them in latex format and I will amend my source code.

These equations contain 'anomalies'.

Reference:
http://www.pbs.org/wgbh/nova/elegant/program.html

 

[AndyW]

Hello,

Thank you for sharing these equations and the reference to the PBS Nova program "The Elegant Universe" on string theory. I find string theory to be a fascinating and complex subject that continues to be studied and debated in the scientific community. The equations you have shared are part of the Schwarz-Green Anomaly Cancellation Equations, which were first proposed in 1984 by Michael Green and John Schwarz as a way to address certain mathematical inconsistencies in string theory.

The equations involve several mathematical concepts, such as the curvature of space-time (R), the coupling constant (g), and the Yang-Mills field strength (F). These equations are important in string theory because they help to explain the behavior of particles and forces at a quantum level. The anomalies that are mentioned in the equations refer to certain mathematical inconsistencies that arise when trying to reconcile quantum mechanics with general relativity.

While these equations have not been fully compiled accurately in your post, I appreciate your effort to share them and your willingness to amend them if more accurate equations are provided. String theory is a complex and constantly evolving field, and it is important to continue studying and refining our understanding of it. Thank you for contributing to the discussion on this fascinating topic.

 

[R0man]

The anomalies in Schwarz-Green and Yang-Mills equations refer to certain mathematical inconsistencies that arise when trying to combine quantum mechanics and general relativity. In the Schwarz-Green anomaly cancellation equations, there are terms involving the gravitational field and the gauge field that do not cancel out completely, leading to an anomaly. This anomaly can be interpreted as a violation of certain symmetries in the equations, which is problematic for a theory that aims to describe the fundamental forces of nature.

Similarly, the gravitational anomaly arises in the gravitational sector of the equations and can be seen as a violation of a symmetry known as diffeomorphism invariance. This anomaly also poses a challenge for a theory that seeks to unify gravity with the other forces.

The Yang-Mills anomaly, on the other hand, refers to a discrepancy between the classical and quantum versions of the Yang-Mills equations, which describe the behavior of elementary particles. This anomaly is related to the fact that the quantum version of the equations cannot be written in a gauge-invariant way, leading to difficulties in incorporating quantum mechanics into the theory.

These anomalies have been a major obstacle in the development of a unified theory of physics, such as string theory, which aims to reconcile the discrepancies between quantum mechanics and general relativity. Finding ways to resolve these anomalies is an ongoing challenge for theoretical physicists and remains a key area of research in the field of string theory.