How can one ensure Lorentz invariance when using an external field in QFT?

In summary: Eugene.OK, the Lagragian should be Lorentz invariant if $phi$ and $J$ are scalr classical fields. Now, when only \phi is quantized, \phi's Lorentz transformation rule is still the same. So, to keep the Lorentz invariance of this "new" Lagragian, I require that $J$ transforms in the same way as $\phi$. Same for translation symmetry... I understand that you cannot find a set of P^{\mu}, J^{\mu\nu} operators to satisfy those commutation relations, since J is not quantized. But in the sense of classical field theory, the Lagragian is still Lorent
  • #1
smileii
3
0
Some friend asked me the following question:

For a real scalar field \phi, assume that H = H_free - \int d^3 x\ J \phi. J(x, t) is just some real number, source, or background field, without second quantization. Now, what is the amplitude \psi(x, t) for finding a particle at time t(before, during, or after source is on/off) at position x? The J(x,t) is nonzero only for finite period of time. And the initial state is vacuum, when t --> -\infty .

This question looks simple. However, I cannot find a solution which satisfies both causality and Lorentz invariance.
 
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  • #2
smileii said:
Some friend asked me the following question:

For a real scalar field \phi, assume that H = H_free - \int d^3 x\ J \phi. J(x, t) is just some real number, source, or background field, without second quantization. Now, what is the amplitude \psi(x, t) for finding a particle at time t(before, during, or after source is on/off) at position x? The J(x,t) is nonzero only for finite period of time. And the initial state is vacuum, when t --> -\infty .

This question looks simple. However, I cannot find a solution which satisfies both causality and Lorentz invariance.

In my opinion, this problem is not correctly formulated. Your Hamiltonian has explicit time dependence, so it describes not an isolated system but a system in an external field. In this case, I am even not sure what is the correct definition of Lorentz invariance. (In the case of isolated systems, the Lorentz invariance means that generators of inertial transformations - the total momentum, angular momentum, Hamiltonian, and boost operators - satisfy commutation relations of the Poincare Lie algebra).

What is the physical situation which your friend tried to model with this Hamiltonian?

Eugene.
 
  • #4
To meopemuk:

I think the external source makes sense in some cases. For example, if we consider the scattering of electron by some electroganetic field, usually, the electron field is quantized, but not the EM field. This half-second- quantization is for calculation simplicity.
 
  • #5
smileii said:
To meopemuk:

I think the external source makes sense in some cases. For example, if we consider the scattering of electron by some electroganetic field, usually, the electron field is quantized, but not the EM field. This half-second- quantization is for calculation simplicity.

I agree that external field is a useful approximation. My question was: how one can guarantee the Lorentz (Poincare) invariance in this approximation? External field defines a preferred frame of reference, then the original question "find a solution which satisfies ... Lorentz invariance." becomes ill-posed. In my opinion.

Eugene.
 
  • #6
meopemuk said:
I agree that external field is a useful approximation. My question was: how one can guarantee the Lorentz (Poincare) invariance in this approximation? External field defines a preferred frame of reference, then the original question "find a solution which satisfies ... Lorentz invariance." becomes ill-posed. In my opinion.

Eugene.

OK, the Lagragian should be Lorentz invariant if $phi$ and $J$ are scalr classical fields. Now, when only \phi is quantized, \phi's Lorentz transformation rule is still the same. So, to keep the Lorent invariance of this "new" Lagragian, I require that $J$ transforms in the same way as $\phi$. Same for translation symmetry... I understand that you cannot find a set of P^{\mu}, J^{\mu\nu} operators to satisfy those commutation relations, since J is not quantized. But in the sense of classical field theory, the Lagragian is still Lorent invariant.

In the exterme case, when $J(x') = \delta^4(x' - y)$, and you detect particle at x, you will expect your answer should be the for of f(x-y) due to translation symmetry. And the amplitude of detection should be still the same when x-> x'. y -> y' in another reference frame. For example, $G(x-y) \theta(x^0 - y^0)$ clearly doesn't satisfy Lorentz invariance. This is how I check "Lorentz symmetry" here.
 

1. How do you detect a particle in Quantum Field Theory (QFT)?

The detection of a particle in QFT is based on the principles of scattering amplitudes and Feynman diagrams. These diagrams represent the interactions between particles and can be used to calculate the probability of a certain particle being created or destroyed during a collision. By analyzing the outcomes of these collisions, scientists can infer the presence of a particle.

2. What is the role of detectors in detecting particles in QFT?

Detectors are crucial in detecting particles in QFT as they are responsible for measuring the properties of particles, such as their energy, momentum, and charge. These measurements are essential in verifying the predictions made by theoretical models and confirming the existence of a new particle.

3. Can particles be detected directly in QFT?

No, particles cannot be detected directly in QFT. This is because, in QFT, particles are described as excitations in a field rather than as discrete objects. Therefore, their properties can only be observed through their interactions with other particles or fields.

4. How do scientists distinguish between different particles in QFT?

In QFT, particles are distinguished by their unique properties, such as mass, spin, and charge. These properties can be measured using detectors and compared to the predicted values from theoretical models. Additionally, the interactions between particles can also provide information about their identity.

5. Is there a limit to the number of particles that can be detected in QFT?

In theory, there is no limit to the number of particles that can be detected in QFT. However, the complexity of detecting multiple particles increases as the number of particles involved in an interaction increases. This is due to the complicated calculations required to analyze the interactions between multiple particles.

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