# How to normalize wave functions in QFT? such as \lambda \phi 4 theory?

• mings6
In summary, the wave function in quantum mechanics is normalized with \int |\phi|^2 dx^3 =1, while the field in quantum field theory is not normalized because it represents the state of the system. However, for states in QFT, normalization is assumed. In the case of solving equations in the same way as the Dirac equation, normalization is needed and the trivial expression of <initial | final> = <\phi|\phi>=1 indicates that the amplitude from \phi to its own is 1 when there is no perturbation. The field in \phi^4 theory is usually not interpreted as a probability density amplitude, so normalization is not necessary. In QFT, the normalization of field operators is defined through equal
mings6
In quantum mechanics, most wave functions are normalized with \int |\phi|^2 dx^3 =1. But I did not see any field in the quantum field theory is normalized. I understand they maybe just plain waves and does not need to be normalized. But in some cases, if we do not expand the field as plain wave, how to normalize them? For instance, in the \lambda \phi 4 theory, the field \phi has the dimension of GeV. Should we use \int |\phi|^4 dx^4 =1 instead of \int |\phi|^2 dx^3 =1 or \int |\phi|^4 dx^3 =1?

The wave function in quantum mechanics represents the state of the system. The field in quantum field theory is not the state, so why do you expect that it be normalized? On the other hand the states in QFT are assumed normalized.

If we want to solve the \lambda \phi 4 equation in the same way as solve the Dirac equation, I think we should normalize the result. So we can say that the trivial expresion of <initial | final> = <\phi|\phi>=1 means, when no perturbation, the amplitude from \phi to its own is 1. For instance, the free particle solution of the Dirac equation is normalized to 1 by a factor \sqrt{m/E}.

Normalization of a field is needed only when that field ITSELF is interpreted as a probability density amplitude. The field \phi in \phi^4 is usually not interpreted in that way.

In QFT the normalization of the field operators are defined via the equal-time commutation relations,

$[\phi(t,\vec{x}),\Pi(t,\vec{y})]=\mathrm{i} \delta^{(3)}(\vec{x}-\vec{y}).$

Here $\Pi$ is the canonical field momentum for $\phi$. In $\phi^4$ theory, it's

$\Pi(x)=\frac{\partial \mathcal{L}}{\partial \dot{\phi}(x)}=\dot{\phi}(x).$

For the asymptotically free states, symbolized by external legs in Feynman diagrams, the states are to be normalized in the usual way to $\delta$ distributions (supposed you have taken account of wave-function renormalization in the external legs, i.e., left out all self-energy insertions in them, see Weinberg, QT of Fields, vol. 1).

## 1. What is the purpose of normalizing wave functions in QFT?

Normalizing wave functions in quantum field theory (QFT) is necessary in order to ensure that the total probability of finding a particle in any region of space is equal to 1. This is a fundamental principle in quantum mechanics, and normalizing wave functions allows for accurate predictions of the behavior of particles in a quantum system.

## 2. How do you normalize a wave function in QFT?

In QFT, wave functions are typically normalized by dividing them by a normalization constant. This constant is found by integrating the square of the wave function over all space and setting it equal to 1. This ensures that the total probability of finding a particle in any region is equal to 1.

## 3. Why is normalization important in QFT?

Normalization is important in QFT because it ensures that the wave function represents a physically meaningful description of a quantum system. Without normalization, the total probability of finding a particle in any region of space may not be accurate, leading to incorrect predictions of particle behavior.

## 4. Is normalization necessary for all wave functions in QFT?

Yes, normalization is necessary for all wave functions in QFT. This is because the normalization condition is a fundamental principle in quantum mechanics and must be satisfied for accurate predictions of particle behavior. However, there may be some cases where wave functions are already normalized, such as in the case of free particles.

## 5. Are there any challenges in normalizing wave functions in QFT?

One challenge in normalizing wave functions in QFT is that the integration over all space can be difficult to perform analytically. This often requires the use of numerical methods, which can be computationally intensive. Additionally, in certain QFT models, the normalization constant may be infinite, which requires the use of regularization techniques to obtain a finite value.

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