Presentation on theme: "From Quantum Mechanics to Lagrangian Densities"— Presentation transcript:
1 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.
2 Summary: Quantum Mechanics The ten minute course in QM.MomentumBecomes anoperatorand useThe Hamiltonianbecomes an operator.
3 Physical interpretation of the wave function. This condition places a strong mathematical condition on the wave function.
4 Note that the Schrodinger equation reflects thisrelationship
5 Quantization arises from placing boundary conditions on the wave function. It is a mathematical result!L = r x P = r x
6 A “toy” model postulate approach to quantum field theory
11 The “negative energy” states arose from This emerges from starting out with.We know the energy of a real particle can’t be < 0.
12 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, ).
13 Following the same derivation used for the A vector we have: Solving the wave equationFollowing the same derivation used for the A vector we have:This is just like the E & Mequation – except for themass term.We try a solution of the form
14 We can see that after taking the partial derivatives there is a condition on the components of k and k0.Note, now the 4-dimensional dot product cannot = 0.The k and k0 (with p= k and p0 = k0 )must satisfy the same conditions as arelativistic particle with rest mass, m.
15 We have the following two linearly independent solutions to the “wave equation”:The most general (complete) solution to the wave equation is
16 The field operator for a neutral, spin =0, particle is creates a singleparticle withmomentump= k andp0 = k0at (r,t)Destroys a singleparticle withmomentump= k andp0 = k0at (r,t)
17 and the Euler-Lagrange equations give F = ma Lagrangians and the Lagrangian DensityRecall that,and the Euler-Lagrange equations give F = maIn quantum field theory, the Euler-Lagrangeequations give the particle wave function!
18 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
19 Note that the Lagrangian density is quadratic in (r,t) and the Lorentz invariance is satisfied by using µ and µ
20 We can apply the Euler Lagrange equations to the above L: This part hasa very simpleresultbutit is hard tocarry out.This part iseasy.
21 Finally, with this Lagrangian density The Euler-Lagrange Equations give the wave equationfor the neutral spin= 0 particle.
22 Summary for neutral (Q=0) scalar (spin = 0) particle, , with mass, m. Lagrangian densitywave equationfield operator