# Time as self-adjoint operator?

• arivero

#### arivero

Gold Member
It has been asked if there exists an operator representing time. I remember some negative answers, but I do not remember the arguments.

In relativistic QM this relates to the non-existence of position operators; we are working with an (t,x,1,x2,x3) vector there. Also I believe to remember some provlem with velocity operators -its eigenvectors been only +c and -c, or so- but I am unsure about if both questions are related.

also, Energy-time indetermination is usedto estimate decay rates of particles. But there an operator is not invoked.

## 1. What is a self-adjoint operator in the context of time?

A self-adjoint operator is a mathematical concept that describes an operator that is equal to its own adjoint. In the context of time, it refers to an operator that represents the passage of time and has the property that its adjoint is also the operator that represents the reverse passage of time.

## 2. How is time represented as a self-adjoint operator in quantum mechanics?

In quantum mechanics, time is represented by the Hamiltonian operator, which is a self-adjoint operator. This operator describes the total energy of a system and governs the time evolution of quantum states.

## 3. What is the significance of time being a self-adjoint operator in quantum mechanics?

The self-adjoint nature of the time operator in quantum mechanics ensures that time evolution is unitary, meaning that the total probability of all possible outcomes remains constant over time. This is a fundamental principle in quantum mechanics.

## 4. Can time be treated as a continuous variable when represented as a self-adjoint operator?

Yes, in quantum mechanics, time is treated as a continuous variable when represented by a self-adjoint operator. This allows for the description of continuous time evolution of quantum systems.

## 5. Are there any limitations or unresolved issues with the concept of time as a self-adjoint operator?

While the concept of time as a self-adjoint operator has been successfully applied in quantum mechanics, it is still a subject of ongoing research and there are some open questions and limitations. For example, the concept of time as a continuous variable conflicts with the discrete nature of time in some theories of quantum gravity.