How Does Time-Reversal Invariance Affect Phi 4 Theory Renormalization?

In summary, the conversation discusses the terms ##\phi_{+}##, ##\phi_{-}##, and ##\phi^4##, and concludes that the vacuum expectation value of the quartic interaction is zero due to time-reversal invariance. Further resources are recommended for further understanding.
  • #1
Diracobama2181
75
2
Homework Statement
Given the energy momentum tensor
##\hat{T}^{\mu \upsilon}## and a Lagrangian with potential ##V=\frac{1}{2}m\phi^2+\lambda\phi^4##,
the quantity ##\bra{\overrightarrow{P'}}T^{\mu v}\ket{\overrightarrow{P}}## should diverge.
How would I go about the 1 loop re-normalization procedure on this term?
Relevant Equations
##\hat{T}^{\mu \upsilon}=\partial^{\mu}\partial^{\upsilon}-g^{\mu \upsilon}L##
##L=\frac{1}{2}\partial_{\mu}\partial^{\mu}-\frac{1}{2}m\phi^2-\lambda\phi^4##
##\phi=\frac{d^3k}{2\omega_k (2\pi)^3}\int(\hat{a}(\overrightarrow{k})e^{-ikx}+\hat{a}^{\dagger}(\overrightarrow{k})e^{ikx}))##
Let ##\phi_{+}=\frac{d^3k}{2\omega_k (2\pi)^3}\int(\hat{a}^{\dagger}(\overrightarrow{k})e^{ikx})##
and ##\phi_{-}=\frac{d^3k}{2\omega_k (2\pi)^3}\int(\hat{a}(\overrightarrow{k})e^{-ikx})##.
Then ##\phi^4=\phi_{1}\phi_{2}\phi_{3}\phi_{4}=(\phi_{1+}+\phi_{1-})(\phi_{2+}+\phi_{2-} )(\phi_{3+}+\phi_{3-}) (\phi_{4+}+\phi_{4-})##
Looking only at the term ##\phi_{1-}\phi_{2-}\phi_{3+}\phi_{4+}##, we find
##\bra{\overrightarrow{P'}}\phi_{1-}\phi_{2-}\phi_{3+}\phi_{4+}\ket{\overrightarrow{P}}## diverges. Unsure where to go from here. Any resources would be helpful. Thank you.
 
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  • #2
A:The quantity you are interested in is the vacuum expectation value of a quartic interaction. This quantity is zero by time-reversal invariance, see e.g. this paper by J. Schwinger.
 

Related to How Does Time-Reversal Invariance Affect Phi 4 Theory Renormalization?

1. What is Phi 4 Theory Renormalization?

Phi 4 Theory Renormalization is a mathematical technique used in quantum field theory to remove infinities that arise in calculations of physical quantities. It involves redefining certain parameters in the theory to account for interactions at different energy scales.

2. Why is Phi 4 Theory Renormalization important?

Phi 4 Theory Renormalization is important because it allows for more accurate predictions in quantum field theory calculations. Without renormalization, infinities would make these calculations impossible. It also helps to reconcile quantum mechanics with special relativity.

3. How does Phi 4 Theory Renormalization work?

Phi 4 Theory Renormalization involves breaking down a physical system into smaller components and then calculating the interactions between these components. The infinities that arise in these calculations are then removed by adjusting certain parameters in the theory, such as the mass and charge of particles.

4. What are the applications of Phi 4 Theory Renormalization?

Phi 4 Theory Renormalization has applications in a variety of fields, including particle physics, condensed matter physics, and cosmology. It is used to study the behavior of particles and their interactions at the smallest scales, and to make predictions about the behavior of matter in extreme conditions.

5. Are there any limitations to Phi 4 Theory Renormalization?

While Phi 4 Theory Renormalization is a powerful tool in quantum field theory, it does have its limitations. It cannot be applied to all physical systems and there are some cases where it may not work. Additionally, the process of renormalization can be complex and time-consuming, making it difficult to apply in certain situations.

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