Useful Fundamental formulas of QFT.

[Mordred]

I am developing a list of fundamental formulas in QFT with a brief description of each to provide some stepping stones to a generalized understanding of QFT treatments and terminology. I invite others to assist in this project. This is an assist not a course. (please describe any new symbols and terms)

 

QFT can be described as a coupling of SR and QM in the non relativistic regime.

 

1) Field :A field is a collection of values assigned to geometric coordinates. Those values can be of any nature and does not count as a substance or medium.

2) As we are dealing with QM we need the simple quantum harmonic oscillator

3) Particle: A field excitation

 

Simple Harmonic Oscillator

[math]\hat{H}=\hbar w(\hat{a}^\dagger\hat{a}+\frac{1}{2})[/math]

the [math]\hat{a}^\dagger[/math] is the creation operator with [math]\hat{a}[/math] being the destruction operator. [math]\hat{H}[/math] is the Hamiltonian operator. The hat accent over each symbol identifies an operator. This formula is of key note as it is applicable to particle creation and annihilation. [math]\hbar[/math] is the Planck constant (also referred to as a quanta of action) more detail later.

 

Heisenberg Uncertainty principle

[math]\Delta\hat{x}\Delta\hat{p}\ge\frac{\hbar}{2}[/math]

 

[math]\hat{x}[/math] is the position operator, [math]\hat{p}[/math] is the momentum operator. Their is also uncertainty between energy and time given by

 

[math]\Delta E\Delta t\ge\frac{\hbar}{2}[/math] please note in the non relativistic regime time is a parameter not an operator.

 

Physical observable's are operators. in order to be a physical observable you require a minima of a quanta of action defined by

 

[math] E=\hbar w[/math]

 

Another key detail from QM is the commutation relations

 

[math][\hat{x}\hat{p}]=\hat{x}\hat{p}-\hat{p}\hat{x}=i\hbar[/math]

 

Now in QM we are taught that the symbols [math]\varphi,\psi[/math] are wave-functions however in QFT we use these symbols to denote fields. Fields can create and destroy particles. As such we effectively upgrade these fields to the status of operators. Which must satisfy the commutation relations

 

[math][\hat{x}\hat{p}]\rightarrow[\hat{\psi}(x,t),\hat{\pi}(y,t)]=i\hbar\delta(x-y)[/math]

[math]\hat{\pi}(y,t)[/math] is another type of field that plays the role of momentum

 

where x and y are two points in space. The above introduces the notion of causality. If two fields are spatially separated they cannot affect one another.

 

Now with fields promoted to operators one wiill wonder what happen to the normal operators of QM. In QM position [math]\hat{x}[/math] is an operator with time as a parameter. However in QFT we demote position to a parameter. Momentum remains an operator.

 

In QFT we often use lessons from classical mechanics to deal with fields in particular the Langrangian

 

[math]L=T-V[/math]

 

The Langrangian is important as it leaves the symmetries such as rotation invariant (same for all observers). The classical path taken by a particle is one that minimizes the action

 

[math]S=\int Ldt[/math]

 

the range of a force is dictated by the mass of the guage boson (force mediator)

[math]\Delta E=mc^2[/math] along with the uncertainty principle to determine how long the particle can exist

[math]\Delta t=\frac{\hbar}{\Delta E}=\frac{\hbar}{m_oc^2}[/math] please note we are using the rest mass (invariant mass) with c being the speed limit

 

[math] velocity=\frac{distance}{time}\Rightarrow\Delta{x}=c\Delta t=\frac{c\hbar}{mc^2}=\frac{\hbar}{mc^2}[/math]

 

from this relation one can see that if the invariant mass (rest mass) m=0 the range of the particle is infinite. Prime example gauge photons for the electromagnetic force.

 

Lets return to [math]L=T-V[/math] where T is the kinetic energy of the particle moving though a potential V using just one dimension x. In the Euler-Langrange we get the following

 

[math]\frac{d}{dt}\frac{\partial L}{\partial\dot{x}}-\frac{\partial L}{\partial x}=0[/math] the dot is differentiating time.

 

Consider a particle of mass m with kinetic energy [math]T=\frac{1}{2}m\dot{x}^2[/math] traveling in one dimension x through potential [math]V(x)[/math]

 

Step 1) Begin by writing down the Langrangian

 

[math]L=\frac{1}{2}m\dot{x}^2-V{x}[/math]

 

next is a derivative of L with respect to [math]\dot{x}[/math] we treat this as an independent variable for example [math]\frac{\partial}{\partial\dot{x}}(\dot{x})^2=2\dot{x}[/math] and [math]\frac{\partial}{\partial\dot{x}}V{x}=0[/math] applying this we get

 

step 2)

[math]\frac{\partial L}{\partial\dot{x}}=\frac{\partial}{\partial\dot{x}}[\frac{1}{2}m\dot{x}^2]=m\dot{x}[/math]

 

which is just mass times velocity. (momentum term)

 

step 3) derive the time derivative of this momentum term.

 

[math]\frac{d}{dt}\frac{\partial L}{\partial\dot{x}}=\frac{d}{dt}m\dot{x}=\dot{m}\dot{x}+m\ddot{x}=m\ddot{x}[/math] we have mass times acceleration

 

Step 4) Now differentiate L with respect to x

 

[math]\frac{\partial L}{\partial x}[\frac{1}{2}m\dot{x}^2]-V(x)=-\frac{\partial V}{\partial x}[/math]

 

Step 5) write the equation to describe the dynamical behavior of our system.

 

[math]\frac{d}{dt}(\frac{\partial L}{\partial\dot{x}}-\frac{\partial L}{\partial x}=0[/math][math]\Rightarrow\frac{d}{dt}[/math][math](\frac{\partial L}{\partial\dot{x}})[/math][math]=\frac{\partial L}{\partial x}\Rightarrow m\ddot{x}=-\frac{\partial V}{\partial x}[/math]

 

recall from classical physics [math]F=-\nabla V[/math] in 1 dimension this becomes [math]F=-\frac{\partial V}{\partial x}[/math] therefore [math]\frac{\partial L}{\partial x}=-\frac{\partial V}{\partial x}=F[/math] we have [math]m\ddot{x}-\frac{\partial V}{\partial x}=F[/math]

Edited July 23, 2017 by Mordred
Updating for new commands latex for math

[StringJunky]

I am developing a list of fundamental formulas in QFT with a brief description of each to provide some stepping stones to a generalized understanding of QFT treatments and terminology. I invite others to assist in this project. This is an assist not a course. (please describe any new symbols and terms)

 

QFT can be described as a coupling of SR and QM in the non relativistic regime.

 

1) Field :A field is a collection of values assigned to geometric coordinates. Those values can be of any nature and does not count as a substance or medium.

2) As we are dealing with QM we need the simple quantum harmonic oscillator

3) Particle: A field excitation

 

Simple Harmonic Oscillator

[latex]\hat{H}=\hbar w(\hat{a}^\dagger\hat{a}+\frac{1}{2})[/latex]

s

the [latex]\hat{a}^\dagger[/latex] is the creation operator with [latex]\hat{a}[/latex] being the destruction operator. [latex]\hat{H}[/latex] is the Hamiltonian operator. The hat accent over each symbol identifies an operator. This formula is of key note as it is applicable to particle creation and annihilation. [latex]\hbar[/latex] is the Planck constant (also referred to as a quanta of action) more detail later.

 

Heisenberg Uncertainty principle

[latex]\Delta\hat{x}\Delta\hat{p}\ge\frac{\hbar}{2}[/latex]

 

[latex]\hat{x}[/latex] is the position operator, [latex]\hat{p}[/latex] is the momentum operator. Their is also uncertainty between energy and time given by

 

[latex]\Delta E\Delta t\ge\frac{\hbar}{2}[/latex] please note in the non relativistic regime time is a parameter not an operator.

 

Physical observable's are operators. in order to be a physical observable you require a minima of a quanta of action defined by

 

[latex] E=\hbar w[/latex]

 

Another key detail from QM is the commutation relations

 

[latex][\hat{x}\hat{p}]=\hat{x}\hat{p}-\hat{p}\hat{x}=i\hbar[/latex]

 

Now in QM we are that the symbols [latex]\varphi,\psi[/latex] are wave-functions however in QFT we use these symbols to denote fields. Fields can create and destroy particles. As such we effectively upgrade these fields to the status of operators. Which must satisfy the commutation relations

 

[latex][\hat{x}\hat{p}]\rightarrow[\hat{\psi}(x,t),\hat{\pi}(y,t)]=i\hbar\delta(x-y)[/latex]

[latex]\phi(y,t)[/latex] is another type of field that plays the role of momentum

 

where x and y are two points in space. The above introduces the notion of causality. If two fields are spatially separated they cannot affect one another.

 

Now with fields promoted to operators one wiill wonder what happen to the normal operators of QM. In QM position [latex]\hat{x}[/latex] is an operator with time as a parameter. However in QFT we demote position to a parameter. Momentum remains an operator.

 

In QFT we often use lessons from classical mechanics to deal with fields in particular the Langrangian

 

[latex]L=V-T[/latex]

 

The Langrangian is important as it leaves the symmetries such as rotation invariant (same for all observers). The classical path taken by a particle is one that minimizes the action

 

[latex]S=\int Ldt[/latex]

 

the range of a force is dictated by the mass of the guage boson (force mediator)

[latex]\Delta E=mc^2[/latex] along with the uncertainty principle to determine how long the particle can exist

[latex]\Delta t=\frac{\hbar}{\Delta E}=\frac{\hbar}{m_oc^2}[/latex] please note we are using the rest mass (invariant mass) with c being the speed limit

 

[latex] velocity=\frac{distance}{time}\Rightarrow\Delta{x}=c\Delta t=\frac{c\hbar}{mc^2}=\frac{\hbar}{mc^2}[/latex]

Cannot understand the red bolded, you've missed words out, I think.

Edited May 20, 2017 by StringJunky

[Mordred]

cross post but thanks for the catch the missing word is taught and its corrected above. I will continue the above tomorrow as its getting late. The next set will getr into the Euler-Langrange in more detail to the action principle. I need time to format my thoughts on that. As the last equation takes 5 steps to solve I have to work out an example to use.

Edited May 20, 2017 by Mordred

[Mordred]

added the details I mentioned to the above.

[KipIngram]

Now in QM we are taught that the symbols [latex]\varphi,\psi[/latex] are wave-functions however in QFT we use these symbols to denote fields. Fields can create and destroy particles. As such we effectively upgrade these fields to the status of operators. Which must satisfy the commutation relations

 

[latex][\hat{x}\hat{p}]\rightarrow[\hat{\psi}(x,t),\hat{\pi}(y,t)]=i\hbar\delta(x-y)[/latex]

[latex]\phi(y,t)[/latex] is another type of field that plays the role of momentum

 

where x and y are two points in space. The above introduces the notion of causality. If two fields are spatially separated they cannot affect one another.

 

The equation has an operator pi in it, but you mention something different in the text - is that a typo?

In step 4 of your Lagrangian discussion I'd say "differentiate" instead of "derive."

Really clean and easy to understand so far - thanks again for the energy you invest around here.

Edited May 22, 2017 by KipIngram

[Mordred]

Yeah its a typo good catch...+1 must have had a blonde moment when typing in the latex...

I like differentiate better will fix up both.

 

edit:Both fixed also repaired the latex mistake in another line.

Edited May 22, 2017 by Mordred

[Mordred]

Original has been reformatted for new latex structure