# Field theory in non-relativistic QM

• Hymne
In summary, the reader is discussing the non-relativistic Hamiltonian and its representation using integral and creation, destruction operators. The wave function is represented as a wave vector in a function space, and the formula given in the conversation is a functional version of this representation. This is similar to representing a vector in 3-space using unit vectors.

#### Hymne

Hello fellow physicists!
I'm currently trying to learn some QFT and the reader gives an introduction by expressing the non-relativistic hamiltonian with integral and creation, destruction operators.

Later he writes:

|Psi, t > = \int d3x1...3xn Psi(x1, ..., xn; t) a+(x1) ... |0>

And i´m not really sure how to think about this. Isn't |Psi, t > the wavevector of Psi(x1, ..., xn; t) ?
Please give me some handwaving or mental words about why this is true..

Hymne said:
|Psi, t > = \int d3x1...3xn Psi(x1, ..., xn; t) a+(x1) ... |0>

Isn't |Psi, t > the wave vector of Psi(x1, ..., xn; t) ?:

$$|x_1,...,x_n>:=a^*(x_1)...a^*(x_n)|0>$$ is a basis vector,
$$\Psi(x_1,...,x_n,t)$$ is the wave function, and
$$\Psi(t)=|\Psi,t>$$ is the wave vector.

In first quantized notation, the wave function is regarded as a wave vector in a function space. Then $$|x_1,...x_n>:=\delta_{x_1}\otimes...\delta_{x_n}$$, where $$\delta_z$$ is the delta function with support at z. Then your formula reduces to a trivial identity.

This is a functional version of the representation of a vector v in 3-space as
$$v=(v_1,v_2,v_3)^T = v_1|1>+v_2|2>+v_3|3>,$$
where |k> is the k-th unit vector.

## 1. What is the basic concept of field theory in non-relativistic quantum mechanics?

Field theory in non-relativistic quantum mechanics is a theoretical framework that combines the principles of classical field theory with the principles of quantum mechanics. It describes the behavior of particles and their interactions in terms of quantum fields, which are mathematical objects that extend throughout space and time.

## 2. How does field theory in non-relativistic quantum mechanics differ from classical field theory?

In classical field theory, particles are treated as point-like objects with definite positions in space and time. However, in field theory in non-relativistic quantum mechanics, particles are described as waves or excitations in a quantum field. This means that their positions and momenta are uncertain and can only be described in terms of probabilities.

## 3. What are the main principles of field theory in non-relativistic quantum mechanics?

The main principles of field theory in non-relativistic quantum mechanics are superposition, uncertainty, and quantization. Superposition means that particles can exist in multiple states at the same time. Uncertainty refers to the inherent unpredictability of a particle's position and momentum. Quantization refers to the fact that certain properties, such as energy levels, can only take on discrete values.

## 4. How is field theory in non-relativistic quantum mechanics used in practical applications?

Field theory in non-relativistic quantum mechanics is used in many practical applications, such as in the development of new materials, electronics, and medical imaging techniques. It is also essential in fields like particle physics and cosmology, where it helps us understand the behavior of subatomic particles and the universe as a whole.

## 5. What are some current challenges and open questions in field theory in non-relativistic quantum mechanics?

One of the main challenges in field theory in non-relativistic quantum mechanics is reconciling it with general relativity, the theory of gravity. This has led to the development of theories like quantum field theory in curved spacetime. Other open questions include the nature of dark matter and dark energy, and how to incorporate them into the framework of field theory in non-relativistic quantum mechanics.