Energy Momentum Tensor in Phi^3 Theory

In summary: In the case of our system, there are three diagrams: one for the energy-momentum tensor, one for the position-momentum tensor, and one for the momentum-position tensor. The first two diagrams are just products of the energy and momentum propagators, while the third diagram is a product of the momentum propagator with the position propagator:$$\langle \Omega | T_{\mu \nu} | \Omega \rangle = \langle \Omega | \partial_\mu \Phi(x) \partial_\nu \Phi(x) - g_{\mu \nu} \mathcal
  • #1
Diracobama2181
75
2
Homework Statement
If I have an energy-momentum tensor
$$\hat{T}^{\mu \nu} (x)=\partial^{\mu}\Phi\partial^{\nu}\Phi-\\g^{\mu \nu}\mathcal{L}$$
,with
$$
\mathcal{L}=(\frac{1}{2}(\partial_{\mu}\Phi)^2-\frac{1}{2}m^2\Phi^2-\frac{\lambda \Phi^3}{3!})
$$
, how would I go about calculating
$$
\bra{ \vec{ p'}} T^{\mu,\nu} \ket{ \vec {p}}
$$?
Relevant Equations
##\ket{\vec{p}}=\hat{a}^{\dagger}(\vec{p})\ket{0}## for a free field with ##[\hat{a}({\vec{k})},\hat{a}^{\dagger}({\vec{k'})}]=2(2\pi)^3\omega_k\delta^3({\vec{k}-\vec{k'}})##
$$ \bra{ \vec{ p'}} T_{\mu,\nu} \ket{ \vec {p}}=\bra{\Omega}\hat{a}(\vec{p'})(\partial^{\mu}\Phi\partial^{\nu} \Phi-g^{\mu \nu}\mathcal{L})\hat{a}^{\dagger}(\vec{p})\ket{\Omega}=\bra{\Omega}(\hat{a}(\vec{p'})\partial^{\mu}\Phi\partial^{\nu} \Phi\hat{a}^{\dagger}(\vec{p})\\ -\hat{a}(\vec{p'})g^{\mu \nu}(\frac{1}{2}(\partial_{\mu}\Phi)^2-\frac{1}{2}m^2\Phi^2-\frac{\lambda \Phi^3}{3!}))\hat{a}^{\dagger}(\vec{p})\ket{\Omega}$$
In this instance, would this imply that, for example, $$\bra{\Omega}(\hat{a}(\vec{p'})(\frac{1}{2}(\partial_{\mu}\Phi)^2-\frac{1}{2}m^2\Phi^2-\frac{\lambda \Phi^3}{3!})\hat{a}^{\dagger}(\vec{p}))\ket{\Omega}=lim_{t \to \infty(1-i\epsilon)}\frac{\bra{0}[T(\hat{a}(\vec{p'})(\frac{1}{2}(\partial_{\mu}\Phi)^2-\frac{1}{2}m^2\Phi^2-\frac{\lambda \Phi^3}{3!})\hat{a}^{\dagger}(\vec{p}))e^{-i\int \lambda \frac{\Phi^3}{3!}dt}]\ket{0}}{\bra{0}T e^{-i\int \lambda \frac{\Phi^3}{3!}dt}\ket{0}}$$?
Also, how would I go about calculating terms like $$\bra{\Omega}\hat{a}(\vec{p'})(\partial^{\mu}\Phi\partial^v \Phi )\hat{a}^{\dagger}(\vec{p})\ket{\Omega}$$ using Feynman diagrams? Thanks in advance.
Note: $$\ket{\Omega}$$ is the ground state in the interacting theory.
 
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  • #2
A:I'm going to give a somewhat abbreviated answer here, because it's probably beyond the scope of this site to provide a full textbook-level explanation.The correct way to calculate matrix elements of the energy-momentum tensor is to perform a Wick contraction; in other words, you need to express the operators inside the expectation value using time-ordered products, and then take the appropriate number of derivatives of the resulting expression with respect to the metric tensor. This approach is explained in some detail in chapter 3 of Peskin and Schroeder's "An Introduction to Quantum Field Theory".In the case at hand, we have an expectation value of the form$$\langle \Omega | T_{\mu \nu} | \Omega \rangle = \langle \Omega | \partial_\mu \Phi \partial_\nu \Phi - g_{\mu \nu} \mathcal{L} | \Omega \rangle$$where $\mathcal{L}$ is the Lagrangian density of the theory. To evaluate this, we use the fact that the vacuum state $|\Omega\rangle$ (or more accurately, the vacuum-to-vacuum expectation value) is annihilated by all annihilation operators, and so$$\langle \Omega | T_{\mu \nu} | \Omega \rangle = \langle \Omega | \partial_\mu \Phi(x) \partial_\nu \Phi(x) - g_{\mu \nu} \mathcal{L}(x) | \Omega \rangle \\= \langle 0 | T \left[ \partial_\mu \Phi(x) \partial_\nu \Phi(x) - g_{\mu \nu} \mathcal{L}(x) \right] | 0 \rangle$$where $T$ denotes the time ordering operator.The second equality follows from the fact that the vacuum-to-vacuum expectation value in an interacting theory is equal to the time ordered expectation value of the same operators in the free theory, which is much easier to calculate.Now, the left-hand side can be expressed as a sum of diagrams, where each diagram is a product of propag
 

Related to Energy Momentum Tensor in Phi^3 Theory

What is the Energy Momentum Tensor in Phi^3 Theory?

The Energy Momentum Tensor in Phi^3 Theory is a mathematical tool used in theoretical physics to describe the distribution of energy and momentum in a system governed by the Phi^3 field theory. It is a tensor quantity that takes into account the energy and momentum of all particles in the system, and is used to calculate important physical quantities such as the stress-energy density and pressure.

How is the Energy Momentum Tensor derived in Phi^3 Theory?

The Energy Momentum Tensor in Phi^3 Theory is derived from the Lagrangian density of the system, which describes the dynamics of the Phi^3 field. By applying the Euler-Lagrange equations, the tensor can be obtained as a conserved current, meaning that its divergence is equal to zero. This reflects the conservation of energy and momentum in the system.

What are the physical implications of the Energy Momentum Tensor in Phi^3 Theory?

The Energy Momentum Tensor in Phi^3 Theory is a crucial tool in understanding the behavior of particles in a system governed by the Phi^3 field theory. It allows us to calculate physical quantities such as the energy and momentum density, the stress-energy tensor, and the pressure, which are important for understanding the dynamics of the system and its interactions with other systems.

How does the Energy Momentum Tensor relate to other concepts in theoretical physics?

The Energy Momentum Tensor in Phi^3 Theory is closely related to other concepts in theoretical physics, such as the Noether theorem, which states that every continuous symmetry in a system has a corresponding conserved current. The tensor is also related to the stress-energy tensor in general relativity, which describes the curvature of spacetime due to the presence of matter and energy.

What are some current research topics related to the Energy Momentum Tensor in Phi^3 Theory?

Some current research topics related to the Energy Momentum Tensor in Phi^3 Theory include its application in cosmology and quantum field theory, as well as its use in understanding the behavior of particles in high-energy collisions. There is also ongoing research on the implications of the tensor in the study of phase transitions and critical phenomena in condensed matter systems.

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