# What is the meaning of uncertainty principle in QTF?

• ndung200790
In summary, Quantum Mechanics and Quantum Field Theory are both theories that describe the behavior of particles and their interactions. However, in QFT, particles are represented by fields, and the uncertainty principle holds for inter-particles rather than single particles. QFT also takes into account the concept of locality and the special properties of fields under Lorentz transformations. The role of generalized coordinates is taken by the field, and momentum is defined just like in systems with a finite number of degrees of freedom. The Fourier components of a quantum field are still operators, and wave functions in QFT are either functions of the field or sums over integrals of momentum states. Measurable values of a quantum field are expectations of the field, and correlation functions are Fourier transforms
ndung200790
In Quantum Mechanics the Heizenbeg principle holds for one particle.In QTF Theory,the creation and annihilation operators corresonding to coefficients of Fourier components of transformation,so it seem that each quantum(particle) has a definite momentum.Then it seem that the uncertainty principle holds for inter-particles(many particles),but not for single particle.If it is true,where is the role of Quantum Mechanics in Quantum Field Theory?
Thank you very much in advance.

It seem that because the Identity Principle,we ought to not pose the ''distinguishing'' between one particle(in QM) and many particles(in QTF Theory).Is that correct?

Did you mean QFT? Quantum Field Theory?

QFT is a theory of fields as physical systems. A field has (an uncountably) infinite number of degrees of freedom labeled by the space-time points $x = (ct, \mathbf{r})$. Then, if you have a Lagrangian density $\mathcal{L}$, you can find the canonical momentum density:
$$\pi(\mathbf{r}, t) = \frac{\partial \mathcal{L}}{\partial (\partial_{t}\varphi(\mathbf{r}, t))}$$
The quantum part of the theory are the canonical commutation relations:
$$\begin{array}{lcl} [\varphi(\mathbf{r}, t), \varphi(\mathbf{r}', t)]_{\mp} & = & 0 \\ [\varphi(\mathbf{r}, t), \pi(\mathbf{r}', t)]_{\mp} & = & i \, \hbar \, \delta^{(3)}(\mathbf{r} - \mathbf{r}') \\ [\pi(\mathbf{r}, t), \pi(\mathbf{r}', t)]_{\mp} & = & 0 \end{array}$$
taking the commutator or anticommutator depending on whether the fields correspond to integer or half-integer spin (see Spin-statistics Theorem). Notice that the field operators entering in the above relations are taken at the same instant in time. In this regard, the formulation is apparently not relativistically invariant, but the requirement for locality and the special properties of the fields under Lorentz transformations actually make it so.

So I guess you mean the space coordinate and momentum in Quantum Field Theory simultaneously having the definite values?Is it correct you mean that?

no, in QFT the role of generalized coordinates is taken by the field ($\varphi(x)$) and the space-time 4-vector x plays the role of a (continuous) label. Momentum is defined just like in mechanics of systems with finite number of degrees of freedom.

I think that even inner the QTF Theory,there are some ''average'' value,e.g the momentum p,the energy epsilon of the particle(quantum).

Then,are there any uncertainty relation between momentum(value) and field(value(not operator))?

particles in qft are identified as specific excitations of the field that carry a certain amount of momentum and energy. These excitations are actually the different Fourier modes that are solutions to the equations of (a non-interacting) qft.

The basic problem in qft is the following: Given a particular excitation prepared in the distant past (corresponding to incoming particles), what is the transition amplitude for a particular arrangement of (not necessarily the same) particles coming out in the distant future. This is an idealized scattering experiment.

The square of the modulus of the transition amplitude, plus some extra factors depending on the phase-space volume give the cross section for the particular process. In this way, QFT is probabilistic because there are many outcomes to a scattering experiment.

Thanks very much for your kind and useful helping.

In quantum field theory, unambiguous meaning can be attached only to spacetime integrals of the idealized field components. Thus when comparing two measurements of field averages over different spacetime regions, we cannot speak unambiguously about the temporal sequence of the measurements and therefore in the discussion of measurement possibilieties in field theory essentially new considerations are involved. In the investigations of Landau and Pierls on the measurability of electromagnetic field quantities, they employed point charges as test bodies and therefore were lead to incorrect conclusions. In fact, only measuring bodies with a charge distribution of finite extent should be considered in the testing of the quantum electrodynamic formalism, since every well-defined statement of this formalism refers to averages of field components over finite spacetime regions.

QFT mean Quantum Field Theory

Then,I think that there is a difference between quantum and particle.The quantum has definite 4-momentum,but the particle may or may not have definite 4-momentum.The only particle may have definite space-time coordinate,but quantum corresponding to a Fourier component of field.May be the quantum notion is ''narrower'' than particle notion.Quantum Mechanics use ''particle-wave'' notion but Quantum Field Theory use ''quantum-wave(or field)'' notion.Is that correct?

It seem that one particle in QTF Theory might be ''involve'' some or many quantum.E.g a definite state of electron might involve some ''quantum''.

In QFT, particles are represented by fields, which can have quantized excitations. I'm not sure what you mean by a distinction between "quantum and particle".

If quantum and particle are the same,why they have the definite 4-momentum which is momentum of Fourier component,because in general a particle in a superposition state has not a certain value of 4-momentum?

I think that the reation and annihilation process of particle(quantum) has probability characteristic.The process changes the Fock states from one to another.If a quantum(particle) has a great momentum,then the fluctuation of momentum in the Fock state ''space'' is large,so following Quantum Mechanics(many body) the coordinate r of particle is definite(Heizenberg principle).Is that correct?
By the way,it seem that the Fourier component of field can not be considered as the wave function of particle(corresponding Fourier component quantum )

By the way,please teach me what is the difference between the value(not operator) of quantum field and wave function?

ndung200790 said:
By the way,it seem that the Fourier component of field can not be considered as the wave function of particle(corresponding Fourier component quantum )

The Fourier components of a quantum field are still operators.

Wave functions in QFT are either (in the functional representation of Phi^4 theory, say) functions of the field Phi, or in the more often used Fock representation sums over N of integrals over N-particle momentum states.

Measurable values of a quantum field are expectations of the field (at particular points in space-time). Also measurable are simple enough correlation functions, which are Fourier transforms of expectations of products of fields.

## 1. What is the uncertainty principle in quantum theory?

The uncertainty principle, also known as the Heisenberg uncertainty principle, is a fundamental principle in quantum mechanics that states that it is impossible to know both the position and momentum of a particle with absolute certainty. This means that the more precisely we know the position of a particle, the less we know about its momentum, and vice versa.

## 2. Why is the uncertainty principle important in quantum mechanics?

The uncertainty principle plays a crucial role in quantum mechanics because it sets a limit on the precision with which we can measure the properties of subatomic particles. It also helps explain the wave-like behavior of particles and the probabilistic nature of quantum systems.

## 3. How does the uncertainty principle relate to the wave-particle duality?

The uncertainty principle is closely related to the wave-particle duality of quantum systems. It states that particles can exhibit both wave-like and particle-like behavior, and the uncertainty principle helps to explain this behavior by showing that it is impossible to know both the position and momentum of a particle with certainty at the same time.

## 4. Can the uncertainty principle be violated?

No, the uncertainty principle is a fundamental principle of quantum mechanics and cannot be violated. It is a consequence of the wave-like nature of particles and the probabilistic nature of quantum systems.

## 5. How does the uncertainty principle impact our understanding of the physical world?

The uncertainty principle challenges our classical understanding of the physical world, where properties of particles are assumed to be determinate. It forces us to accept the probabilistic nature of the quantum world and has led to groundbreaking discoveries and technologies, such as quantum computing and cryptography.

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