Single Particle Expectation of Energy Momentum Tensor

In summary: Q})\hat{a}^\dagger(\overrightarrow{Q'})\ket{0}\tag{4} \end{align*}The second term is just the annihilation operator, so it's just\begin{align*}\bra{\overrightarrow{P'}}\partial^{v}\Phi(0)\partial^{\mu}\Phi(0)\ket{\overrightarrow{P}}&=\bra{0}\int\frac{d^
  • #1
Diracobama2181
75
2
Homework Statement
Currently trying to explicitly calculate the following for a non interacting spin 0 field

$$\bra{\overrightarrow{P'}}\hat{T}_{\mu v}\ket{\overrightarrow{P}}
$$
where
$$\hat{T}_{\mu v}=\partial^{\mu}\Phi \partial^{v}\Phi-g^{\mu v}L$$
Relevant Equations
For this setup,
$$L=\frac{1}{2}\partial_{\mu}\Phi \partial^{\mu}\Phi-\frac{1}{2}m^2\Phi^2 $$
$$\Phi=\int\frac{d^3k}{2\omega_k (2\pi)^3}(\hat{a}(\overrightarrow{k})e^{-ikx}+\hat{a}^{\dagger}(\overrightarrow{k})e^{ikx}))$$
$$\ket{\overrightarrow{P}}=a^{\dagger}\ket{0}$$
and $$[a(\overrightarrow{k'}),a^{\dagger}(\overrightarrow{k})]=2\omega_k (2\pi)^3 \delta^3(\overrightarrow{k'}-\overrightarrow{k})$$
$$\hat{T}_{\mu v}(x)=e^{i\hat{P}x}\hat{T}_{\mu v}(0)e^{-i\hat{P}x}$$,
so $$\bra{\overrightarrow{P'}}\hat{T}_{\mu v}(x)\ket{\overrightarrow{P}}=e^{iP'x}\bra{\overrightarrow{P'}}\hat{T}_{\mu v}(0)\ket{\overrightarrow{P}}e^{-i\hat{P}x}$$
Now,
$$\partial^{\mu}\Phi=\int\frac{d^3 k_1}{2\omega_{k_1} (2\pi)^3}(-ik_1^{\mu}\hat{a}(\overrightarrow{k_1})e^{-ik_1 x}+i k_1^{\mu}\hat{a}^{\dagger}(\overrightarrow{k_1})e^{ik_1 x}))$$
and $$\partial^{v}\Phi=\int\frac{d^3 k_2}{2\omega_{k_2}(2\pi)^3}(-ik_2^{v}\hat{a}(\overrightarrow{k_2})e^{-ik_2 x}+ik_2^{v}\hat{a}^{\dagger}(\overrightarrow{k_2})e^{ik_2 x}))$$
so
$$\bra{\overrightarrow{P'}}\partial^{\mu}\Phi(0)\partial^{v}\Phi(0)\ket{\overrightarrow{P}}=\bra{0}\int\frac{d^3 k_1 d^3 k_2}{4\omega_{k_1}\omega_{k_2} (2\pi)^6}(-k_1^{\mu}k_2^{v}\hat{a'}\hat{a}_{k_1}\hat{a}_{k_2}\hat{a}^{\dagger}+k_1^{\mu}k_2^{v}\hat{a'}\hat{a}_{k_1}^{\dagger}\hat{a}_{k_2}\hat{a}^{\dagger}
+k_1^{\mu}k_2^{v}\hat{a'}\hat{a}_{k_1}\hat{a}_{k_2}\hat{a}^{\dagger}-k_1^{\mu}k_2^{v}\hat{a'}\hat{a}_{k_1}^{\dagger}\hat{a}_{k_2}^{\dagger}\hat{a}^{\dagger})\ket{0}
=k^{\mu}k^{v'}+k^{\mu'}k^{v}+\int d^3 k_1 d^3 k_2 k_1^{\mu}k_2^{v}\delta^3(\overrightarrow{k_1}-
\overrightarrow{k_2})\delta^3(\overrightarrow{k'}-\overrightarrow{k})$$
Then, using
$$\Phi^2(0)=\int \frac{d^3k_1 d^3k_1}{4\omega_{k_1}omega_{k_2} (2\pi)^3}(\hat{a_{k_1}}\hat{a_{k_2}}+\hat{a_{k_1}}\hat{a_{k_2}}^{\dagger}+\hat{a_{k_1}}^{\dagger}\hat{a_{k_2}}+\hat{a_{k_1}}^{\dagger}\hat{a_{k_2}}^{\dagger})$$
and
$$\bra{\overrightarrow{P'}}\Phi^2(0)\ket{\overrightarrow{P}}=\bra{0}\int\frac{d^3k_1 d^3k_1}{4\omega_{k_1}\omega_{k_2} (2\pi)^6}(4(2\pi)^6\omega_{k_1}\omega_{k_2}\delta^3(\overrightarrow{k_1}-\overrightarrow{k_2})\delta^3(\overrightarrow{k'}-\overrightarrow{k})+4(2\pi)^6\omega_{k_1}\omega_{k_2}\delta^3(\overrightarrow{k_1}-\overrightarrow{k})\delta^3(\overrightarrow{k'}-\overrightarrow{k_2})+4(2\pi)^6\omega_{k_1}\omega_{k_2}\delta^3(\overrightarrow{k_2}-\overrightarrow{k})\delta^3(\overrightarrow{k'}-\overrightarrow{k_1}))\ket{0}=2+\int d^3 k_1 d^3 k_1\delta^3(\overrightarrow{k_2}-\overrightarrow{k_1})\delta^3(\overrightarrow{k'}-\overrightarrow{k}))$$
Do these calculations seem correct so far? And if so, how do I go about renormalizing the equations to get rid of the divergences. Thank you.
 
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  • #2
A:You're on the right track, but there's still some cleaning up that needs to be done. First you need to write the expression for $\hat{T}_{\mu v}$ in terms of the creation and annihilation operators:$$\hat{T}_{\mu v}(x)=\partial_\mu\Phi(x)\partial_v\Phi(x)-\eta_{\mu v}\Phi^2(x).$$Now you can calculate each term separately. For the first term, we have\begin{align*}\bra{\overrightarrow{P'}}\partial^\mu\Phi(0)\partial^v\Phi(0)\ket{\overrightarrow{P}}&=\int\frac{d^3k_1d^3k_2}{4\omega_{k_1}\omega_{k_2}(2\pi)^6}(-ik_1^\mu k_2^v\bra{0}\hat{a}(\overrightarrow{k_1})\hat{a}(\overrightarrow{k_2})\hat{a}^\dagger(\overrightarrow{P})\hat{a}^\dagger(\overrightarrow{P'})\ket{0}\tag{1}\\&+ik_1^\mu k_2^v\bra{0}\hat{a}^\dagger(\overrightarrow{k_1})\hat{a}(\overrightarrow{k_2})\hat{a}^\dagger(\overrightarrow{P})\hat{a}^\dagger(\overrightarrow{P'})\ket{0}\tag{2}\\&-ik_1^\mu k_2^v\bra{0}\hat{a}(\overrightarrow{k_1})\hat{a}^\dagger(\overrightarrow{k_2})\hat{a}^\dagger(\overrightarrow{P})\hat{a}^\dagger(\overrightarrow{P'})\ket{0}\tag{3}\\&+ik_1^\mu k_2^v\bra{
 

Related to Single Particle Expectation of Energy Momentum Tensor

1. What is the Single Particle Expectation of Energy Momentum Tensor?

The Single Particle Expectation of Energy Momentum Tensor is a mathematical quantity that describes the average energy and momentum distribution of a single particle in a quantum mechanical system. It is a fundamental concept in quantum mechanics and is used to understand the behavior of particles at the microscopic level.

2. How is the Single Particle Expectation of Energy Momentum Tensor calculated?

The Single Particle Expectation of Energy Momentum Tensor is calculated using the Schrödinger equation, which is a mathematical equation that describes the time evolution of a quantum mechanical system. The equation takes into account the energy and momentum of the particle and their interactions with the surrounding environment.

3. What is the significance of the Single Particle Expectation of Energy Momentum Tensor?

The Single Particle Expectation of Energy Momentum Tensor is significant because it provides information about the quantum state of a single particle, such as its energy and momentum distribution. This information can be used to make predictions about the behavior of the particle in various physical situations.

4. How does the Single Particle Expectation of Energy Momentum Tensor relate to other quantum mechanical concepts?

The Single Particle Expectation of Energy Momentum Tensor is closely related to other quantum mechanical concepts, such as the wave function and the Hamiltonian operator. It is also related to the uncertainty principle, which states that the more precisely the energy and momentum of a particle are known, the less precisely its position can be known.

5. What are some real-world applications of the Single Particle Expectation of Energy Momentum Tensor?

The Single Particle Expectation of Energy Momentum Tensor has many applications in physics, including understanding the behavior of particles in quantum field theory, studying the properties of materials at the atomic level, and predicting the behavior of particles in particle accelerators. It is also used in the development of new technologies, such as quantum computing and quantum cryptography.

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