Phi 4 Theory Propagator Question

In summary: T}\left[\mathcal{L}_{int}(\tau_1)\mathcal{L}_{int}(\tau_2)\right]=\frac{1}{2}\int_{t_0}^{t}d\tau_1\int_{t_0}^{t}d\tau_2\int d^3x_1\int d^3x_2\mathcal{T}\left[\mathcal{H}_{int}(\vec{x}_1,\tau_1)\mathcal{H}_{int}(\vec{x}_2,\tau_2)\right]$$Using these expansions, we can now rewrite the expression for $\Phi(\vec{x},t)$
  • #1
Homework Statement
Find $$\bra{\Omega}\partial^{\mu}\Phi\partial^{\nu}\Phi\ket{\Omega}$$ in $$\Phi^4$$ theory.
Relevant Equations
$$\mathcal{L}=\frac{1}{2}(\partial_{\mu}\Phi)^2-\frac{1}{2}m^2\Phi^2-\lambda\frac{\Phi^4}{4!}$$
I know in the Heisenburg picture,
$$\Phi(\vec{x},t)=U^{\dagger}(t,t_0)\Phi_{0}(\vec{x},t)U(t,t_0)$$
where $$\Phi_{0}$$ is the free field solution, and
$$U(t,t_0)=T(e^{i\int d^4x \mathcal{L_{int}}})$$. Is there a way I could solve this using contractions or Feynman diagrams?
Because otherwise, it would appear I would have to solve this by taking the derivative of $$U^{\dagger}(t,t_0)\Phi_{0}(\vec{x},t)U(t,t_0)$$
and this would give $$\partial^{\mu}\Phi(\vec{x},t)=\partial{\mu}U^{\dagger}(t,t_0)\Phi_{0}(\vec{x},t)U(t,t_0)+U^{\dagger}(t,t_0)\partial^{\mu}\Phi_{0}(\vec{x},t)U(t,t_0)+U^{\dagger}(t,t_0)\Phi_{0}(\vec{x},t)\partial^{\mu}U(t,t_0)$$
which would mean I would have to calculate
$$\partial^{\mu}U(t,t_0)=T(\partial^{v}(i\int d^4x \mathcal{L}_{int}))U(t,t_0)$$
which would make this calculation messy. Thanks in advance.
 
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  • #2

Thank you for your question. Yes, there is a way to solve this using contractions or Feynman diagrams. In fact, this is the standard way to approach quantum field theory calculations.

First, let's rewrite the expression in terms of creation and annihilation operators:
$$\Phi(\vec{x},t)=U^{\dagger}(t,t_0)\Phi_{0}(\vec{x},t)U(t,t_0)=U^{\dagger}(t,t_0)\sum_{\vec{k},\lambda}a_{\vec{k},\lambda}e^{i(\vec{k}\cdot\vec{x}-\omega_{\vec{k}}t)}+a^{\dagger}_{\vec{k},\lambda}e^{-i(\vec{k}\cdot\vec{x}-\omega_{\vec{k}}t)}U(t,t_0)$$

Next, let's expand the time-evolution operator using the Dyson series:
$$U(t,t_0)=\mathcal{T}\left[e^{i\int_{t_0}^{t}d\tau \mathcal{L}_{int}(\tau)}\right]=1+i\int_{t_0}^{t}d\tau\mathcal{L}_{int}(\tau)+\frac{i^2}{2!}\int_{t_0}^{t}d\tau_1\int_{t_0}^{t}d\tau_2\mathcal{T}\left[\mathcal{L}_{int}(\tau_1)\mathcal{L}_{int}(\tau_2)\right]+\cdots$$

Using the definition of the time-ordered product, we can write the second term in the series as:
$$\int_{t_0}^{t}d\tau\mathcal{L}_{int}(\tau)=\int_{t_0}^{t}d\tau\int d^3x \mathcal{H}_{int}(\vec{x},\tau)$$
where $\mathcal{H}_{int}(\vec{x},\tau)$ is the interaction Hamiltonian density.

Using the same procedure, we can expand the third term in the series as:
$$\int_{t_0}^{t}d\tau_1\int_{t_0}^{t}d\tau_
 

1. What is the Phi 4 Theory Propagator?

The Phi 4 Theory Propagator is a mathematical concept used in quantum field theory to describe the propagation of particles and their interactions. It is a fundamental tool for understanding the behavior of particles at the subatomic level.

2. How is the Phi 4 Theory Propagator calculated?

The Phi 4 Theory Propagator is calculated using a mathematical formula that takes into account the mass and energy of the particles involved, as well as their interactions with each other. This formula is derived from the principles of quantum mechanics and is constantly being refined and improved by scientists.

3. What is the significance of the Phi 4 Theory Propagator?

The Phi 4 Theory Propagator is significant because it allows scientists to make predictions and calculations about the behavior of particles and their interactions. It has been used in various fields of physics, including particle physics, condensed matter physics, and cosmology.

4. How does the Phi 4 Theory Propagator relate to other theories?

The Phi 4 Theory Propagator is a part of the Standard Model of particle physics, which is the most widely accepted theory for explaining the behavior of subatomic particles. It also has connections to other theories such as quantum electrodynamics and general relativity.

5. What are some current research topics related to the Phi 4 Theory Propagator?

Some current research topics related to the Phi 4 Theory Propagator include its application in studying the properties of dark matter, its role in understanding the behavior of particles in extreme conditions such as in the early universe or in black holes, and its connection to other areas of physics such as quantum gravity and string theory.

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