From Quantum Mechanics to Lagrangian Densities Just as there is no derivation of quantum mechanics from classical mechanics, there is no derivation of.

From Quantum Mechanics to Lagrangian Densities

Just as there is no “derivation” of quantum mechanics from classical mechanics, there is no derivation of relativistic field theory from quantum mechanics. The “route” from one to the other is based on physically reasonable postulates and the imposition of Lorentz invariance and relativistic kinematics. The final “theory” is a model whose survival depends absolutely on its success in producing “numbers” which agree with experiment.

Summary: Quantum Mechanics

MomentumBecomes anoperator

The ten minute course in QM.

and use

The Hamiltonianbecomes an operator.

This condition places a strong mathematical condition on the wave function.

Physical interpretation of the wave function.

Note that the Schrodingerequation reflects thisrelationship

Quantization arises from placing boundary conditions onthe wave function. It is a mathematical result!

L = r x P = r x

A “toy” model postulate approach to quantum field theory

- -+

- p

p

-

Note that *(r,t) (r,t) does not represent the probabilityper unit volume density of the particle being at (r,t).

- -

The resulting “wave equation”:

The “negative energy” states arose from

This emerges from starting out with

We know the energy of a real particle can’t be < 0.

.

Suppose the mass, m, is zero:

This is the same equation we derived from Maxwell’s equations forthe A vector (except, of course, above we have a scalar, ).

Following the same derivation used for the A vector we have:

We try a solution of the form

Solving the wave equation

This is just like the E & Mequation – except for the mass term.

We can see that after taking the partial derivatives there is a conditionon the components of k and k0.

The k and k0 (with p= k and p0 = k0 ) must satisfy the same conditions as a relativistic particle with rest mass, m.

Note, now the 4-dimensional dot product cannot = 0.

We have the following two linearly independent solutions to the “wave equation”:

The most general (complete) solution to the wave equation is

The field operator for a neutral, spin =0, particle is

creates a singleparticle withmomentump= k and p0 = k0

at (r,t)

Destroys a singleparticle withmomentump= k and p0 = k0

at (r,t)

In quantum field theory, the Euler-Lagrange equations give the particle wave function!

Lagrangians and the Lagrangian Density

Recall that,

and the Euler-Lagrange equations give F = ma

This calls for a different kind of “Lagrangian” -- not like the one used in classical or quantum mechanics. So, we have another postulate, defining what is meant by a “Lagrangian” – called a Lagrangian density.

d/dt in the classical theory

Note that the Lagrangian density is quadratic in (r,t) and the Lorentz invariance is satisfied by using µ and µ

We can apply the Euler Lagrange equations to the above L:

This part iseasy.

This part hasa very simpleresult but it is hard tocarry out.

Finally,

give the wave equationfor the neutral spin= 0 particle.

The Euler-LagrangeEquations

with this Lagrangiandensity

Summary for neutral (Q=0) scalar (spin = 0) particle, , with mass, m.

Lagrangian density

wave equation

field operator

Charged (q = ±e) scalar (spin =0) particle with mass, m

8 termscancel

charged scalar particle

and the Euler-Lagrange equation

From the Lagrangian density

we can derive the wave equation

creates positively charged particle with momentump= k and p0 = k0 at (r,t)

destroys negatively charged particle with momentump= k and p0 = k0 at (r,t)

creates negatively charged particle with momentump= k and p0 = k0 at (r,t)

destroys negatively charged particle with momentump= k and p0 = k0 at (r,t)

The scalar field (which represents a boson) must alsosatisfy a special boson commutation property:

Creation andannihilationoperators withthe same k don’tcommute.

Everything elsecommutes.

Example of how the commutation relation is used

We will use this when we calculate the charge of a particle.Later we will find that fermions satisfy a different commutationrelation.

What follows is for the graduate students.

For the graduate students:

For the undergraduates: You can just remember this

Before taking the partial derivative, it is helpful to rewrite the L.

Let = inthis summation,so it does not become confused with the indexof /x --used later.

For the undergraduates: You can just remember this