Lorentz invariant theory, irreducible representations

In summary, a state in a Lorentz invariant theory in d dimensions can be viewed as an element of a vector space that is acted upon by representations of the subgroup SO(1,d-1) which leaves its momentum unchanged. This means that a state can be classified into different irreducible representations based on the eigenvalues of certain Casimir invariants. In physics, these subspaces are referred to as representations, while in mathematics, they are seen as acting on finite-dimensional vector spaces.
  • #1
physicus
55
3
"In a Lorentz invariant theory in d dimensions a state forms an irreducible representation under the subgroups of [itex]SO(1,d-1)[/itex] that leaves its momentum invariant."

I want to understand that statement. I don't see how I should interpret a state as representation of a group. I have learned that representations of groups are maps which assign a linear transformation between vector spaces (or matrix) to each group element. But why should a state correspond to such a mapping? I am completely lost here.

I would be happy if someone could clear this up for me.
 
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  • #2
Yet another example of physicists mixing up the language. States in a QFT are elements of the vector space that is acted upon by representations of some group. For example, scalars are acted upon by the trivial representation, spinors are acted upon by the spin-1/2 representation, etc.
 
  • #3
physicus, it means that there is an infinite-dim., reducible rep. of the Poincare group defined by a family of unitary operators U[θa] (where θa are the group parameters) acting on the full Hilbert space H, which can be decomposed into irred. reps., corresponding to subspaces of H which are classified w.r.t. to their Casimir invariants PαPα and WαWα (with eigenvalues m² and m²s(s+1) in the massive case m²>0; refer to Wigner's classification for more details).

In theoretical physics one often says that such a (finite dim.) subspace of H is a rep., whereas is mathematics one says that a rep. acts on a (finite dim.) vector space.
 

1. What is the significance of Lorentz invariance in physics?

Lorentz invariance is a fundamental principle in physics which states that the laws of physics should remain the same for all observers moving at a constant velocity. This is a crucial concept in understanding the behavior of particles and fields in special relativity. It also plays a key role in the development of theories like quantum field theory and the standard model of particle physics.

2. What is a representation in the context of Lorentz invariant theory?

In physics, a representation refers to a mathematical framework used to describe the behavior of physical systems. In the context of Lorentz invariant theory, representations are used to classify the different types of particles and fields according to their transformation properties under the Lorentz group. This allows us to understand how particles and fields behave under different conditions and interactions.

3. How are irreducible representations related to Lorentz invariance?

Irreducible representations are a special type of representation that cannot be broken down into smaller parts. In the context of Lorentz invariant theory, irreducible representations are particularly important because they represent the fundamental building blocks of the theory. This means that all other representations can be constructed using combinations of these irreducible representations, making them essential for understanding the behavior of particles and fields in a Lorentz invariant framework.

4. What is the role of group theory in Lorentz invariant theory?

Group theory is a mathematical tool used to study symmetries in physical systems. In the context of Lorentz invariant theory, group theory is used to describe the symmetries of the laws of physics under the Lorentz transformation. This allows us to identify the different types of particles and fields and understand how they interact with each other. Group theory also helps us to construct and manipulate representations, making it an essential tool in the development of Lorentz invariant theories.

5. How does Lorentz invariant theory relate to other theories in physics?

Lorentz invariant theory is a fundamental concept in modern physics and has connections to many other theories, including special relativity, quantum mechanics, and quantum field theory. It provides a framework for understanding the behavior of particles and fields in a relativistic setting and has been essential in the development of the standard model of particle physics. Lorentz invariance is also a key principle in the search for a unified theory that can explain all fundamental interactions in the universe.

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