Extending a linear representation by an anti-linear operator

In summary, the conversation discusses the representation of the O(1,3) group and its extensions by adding in discrete transformations. It is mentioned that while SO(1,3) and P are represented as linear operators, T is represented as anti-linear. The concept of adding T as CU, where U is unitary and C is the complex conjugation operator, is also explored. The conversation then delves into the mathematical framework and the question of whether the extended representation is of a bigger group. References and sources are also mentioned.
  • #1
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When we consider representation of the O(1,3) we divide the representation up into the connected part of SO(1,3) which we then extend by adding in the discrete transformations P and T. However, while (as far as I know) we always demand that the SO(1,3) and P be represented as linear operators, we demand that T be anti-linear.

If I start with some representation of SO(1,3) and I extended it by including T. Since its anti linear and anti unitary it can be written as T = CU where U is unitary and C is the complex conjugation operator. Suppose U were already in SO(1,3), then while I say I am adding T really the only new object I am adding is C. So suppose I have any representation and I extend by C, what is this new object?

I am just looking for the mathematical framework in which to approach this. Once I add T it isn't a linear representation anymore, I see that we have some operators which are homomorphic to O(1,3) but this isn't a representation as I recognize it. So what is this?

While I am framing all this in the context of SO(1,3), it's only because that is how it arose for me. Really, I'm interested in the general question. Suppose I had some linear representation of some group which I then extend by the complex conjugation operator, thereby allowing anti-linear operators. Is this a representation of some bigger group?Any references, sources, etc would be really appreciated.
 
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  • #2
I usually start with the definitions. A representation of ##O(1,3)## is a group homomorphism into ##\operatorname{GL}(V)##. As the orthogonal group is already a matrix group, we get one representation for free.

You could also have a look into:
https://www.amazon.com/dp/0387901086/?tag=pfamazon01-20
 

What is a linear representation?

A linear representation is a mathematical mapping of a given set of objects or structures to another set of objects or structures in a way that preserves the underlying linear structure. This means that the operations of addition and scalar multiplication are preserved in the mapping.

What is an anti-linear operator?

An anti-linear operator is a mathematical function that preserves the structure of a linear space but reverses the order of multiplication. This means that the operator maps the product of two vectors to the product of the operator applied to the individual vectors in the reverse order.

Why is it important to extend a linear representation by an anti-linear operator?

Extending a linear representation by an anti-linear operator allows for a more complete and accurate description of the underlying structure of a system. It can also provide a more efficient and elegant way of representing complex mathematical systems.

What are some examples of extending a linear representation by an anti-linear operator?

One example is in quantum mechanics, where the wave function is a linear representation of the state of a system, but the complex conjugate of the wave function is an anti-linear operator that is used to calculate the probability of finding a particle in a certain state. Another example is in the theory of relativity, where the Lorentz transformation is a linear representation of the relationship between two frames of reference, but the complex conjugate of the Lorentz transformation is an anti-linear operator that is used to transform equations between frames.

What are the potential challenges when extending a linear representation by an anti-linear operator?

One challenge is understanding the underlying mathematical properties and how they are preserved or reversed by the anti-linear operator. Another challenge is finding the right anti-linear operator to use, as there can be multiple possible choices that may give different results. It is also important to ensure that the extended representation is consistent and does not introduce contradictions or inconsistencies in the system.

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