Eigenstates of ##\phi^4## theory

In summary, the conversation discusses the organization of eigenstates in the nonperturbative regime of the ##\phi^4## theory in QFT, including the potential existence of bound states in different dimensions and the relationship between the energy of a multiparticle eigenstate and its constituent particle momenta. The speaker also mentions the lack of information available on the inner product of the non-interacting Klein-Gordon vacuum state with the interacting vacuum state, and raises questions about the definition of ground states in this context. Both parties express frustration with the lack of concrete results in this area.
  • #1
HomogenousCow
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What is known about the eigenstates of the ##\phi^4## theory in QFT? Is there an informal understanding of how these states are organized in the nonperturbative regime? For example, are there known to be any bound states in any dimensions? How does the energy of a multiparticle eigenstate (if anything like this exists) depend on its constituent particle momenta? I realize that there are probably no rigorous answers to these questions, however I would like some kind of intuition as this issue has been bothering me for years, thank you.
 
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  • #2
I tried to find if anything has been written even about the inner product of the non-interacting Klein-Gordon vacuum state with the vacuum of the system with ##\phi^4## interaction turned on, but couldn't find anything. I.e., "how many % of the interacting vacuum is the same as the non-interacting one". Not sure if the two ground states are even defined in the same ##\mathcal{H}##.
 
  • #3
hilbert2 said:
I tried to find if anything has been written even about the inner product of the non-interacting Klein-Gordon vacuum state with the vacuum of the system with ##\phi^4## interaction turned on, but couldn't find anything. I.e., "how many % of the interacting vacuum is the same as the non-interacting one". Not sure if the two ground states are even defined in the same ##\mathcal{H}##.

Yes I've had the same experience looking for information on non-perturbative ##\phi^4## theory, the mathematicians claim that they've formulated the theory in ##d<4## dimensions but I've never seen a single result that it looks like it has anything to do with physics.
 

Related to Eigenstates of ##\phi^4## theory

1. What is an eigenstate in ##\phi^4## theory?

An eigenstate in ##\phi^4## theory is a state in which the field operator ##\hat{\phi}## acts as a scalar multiple of itself. In other words, the state is an eigenvector of the field operator with a corresponding eigenvalue. This means that the field operator acting on the state will return the same state multiplied by a constant factor.

2. How do eigenstates relate to the energy spectrum of ##\phi^4## theory?

Eigenstates are directly related to the energy spectrum of ##\phi^4## theory. Each eigenstate has a corresponding energy eigenvalue, which represents the energy of the state. The energy spectrum of ##\phi^4## theory is made up of all the possible energy eigenvalues that can be obtained by acting the field operator on an eigenstate.

3. Can eigenstates be in a superposition?

Yes, eigenstates can be in a superposition. In fact, most states in quantum mechanics are superpositions of eigenstates. This means that a state can be a linear combination of multiple eigenstates, each with its own eigenvalue. This allows for a more complex and varied description of the state.

4. How are eigenstates used in solving the ##\phi^4## theory equations of motion?

Eigenstates are used in solving the equations of motion in ##\phi^4## theory by providing a basis for the field operator. By expanding the field operator in terms of eigenstates, the equations of motion can be rewritten as a set of coupled differential equations, which can then be solved to find the time evolution of the system.

5. Are there any physical implications of eigenstates in ##\phi^4## theory?

Yes, there are physical implications of eigenstates in ##\phi^4## theory. The energy eigenvalues of the eigenstates correspond to the energy levels of the system, and the eigenstates themselves represent the possible states that the system can be in. These eigenstates and energy levels play a crucial role in understanding the behavior and dynamics of the system.

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