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Sec. 1 Classical Free Field Theory 1. The theory to describe particle physics needs to combine the principles of both Quantum Mechanics and Special Relativity. The usual framework of QM by Schrodinger Wave Equation is non-relativistic and hence doesn’t work in this case. In this formalism, the space position is treated as an operator x̂ , while time is treated as a parameter t, ie a number. It doesn’t allow a Lorentz symmetry that mixes time and space. 2. However, Special Relativity was born from the classical theory of Electromagnetic Field. So a field theory is much easier to be made Lorentz invariant. So let’s speculate that the theory to describe fundamental particles is a field theory but quantized. In the following, first, we work out the simplest classical field theory-scalar field ( x , t ) in the next three pages from Peskin’s book on quantum field theory. 3. A scalar field ( x , t ) is a number function of space and time. Note that space and time are treated equally as parameters. Its dynamics can be determined from its action, or Lagrangian. The beginning of section 2-2 on P.2 describe how to derive its equation of motion, also called Euler Equation (2.3) by the least action principle from the Lagrangian L. This formalism, just like that of particles, can be also rewritten in terms of Hamiltonian. It is done on P.3 following (2.3) 4. For free field, its wave equation is linear so that superposition principle applies. For linear wave equation, the Lagrangian must be quadratic in the field ( x , t ) . The Lagrangian must also be a Lorentz scalar so that the theory is Lorentz invariant. Add one more requirement that the Lagrangian contains at most two space-time derivatives. We can conclude the most general Lagrangian is written as in Eq. (2.6) on P.3. Derive its Euler Eq. and we get (2.7). It’s called Klein-Gorden Equation. 5. To solve this equation classically, we employ the plane wave solutions as e ik x . The general solution is linear combinations of the plane wave solutions, as at the bottom of P.7. From the 1 derivation on P.7, you can find the dispersion relation between angular wavenumber k and angular frequency ω or equivalently the relation between momentum and energy. Look it’s exactly the relation for a relativistic particle! This indicate a close relation between particle and field. The strange thing is that there is a positive solution energy plus a negative energy solution. Why it’s so will become clear when the field is quantized. Note that the integration is Lorenz Invariant. The Lorenz invariance of the Lagrangian density will guarantee the Lorenz invariance of Action and hence EOM. 2 3 4 5 6 To solve the KG Equation, we first expand the field in terms of a Fourier Series: The time dependence can be determined by plugging the series into KG Eq. 2 Every Fourier Component behaves like a SHO with ω: p E p p m2 . KG Field is just a collection of SHO’s. Each SHO is characterized by its k or p “momentum”. The frequency ω or “energy” of the SHO is just that of a relativistic particle with mass m. A general solution is a linear superposition of all plane waves. d3p d3p iEt ip x e c p eiEtipx a p 3 3 2 2 3 d p d 3 p' iEt ip x e c p ' eiEtip'x a p 3 3 2 2 ( x) Here we changed the variable: p' p . But in the integration, the variable is just a dummy and hence can be renamed as p: ( x) d3p a p e d3p a p e ip x b p e ip x 2 3 2 3 iEt ip x d 3 p' 2 3 b p' e iEtip ' x The solution of KG Equation can be written as: d3p a p eipx bp eipx x 3 2 2 with E p p m2 . If the field is real, . Hence bp a p . The real solution of KG Equation can be written as: d3p x a p e ipx a p e ipx 3 2 7 Sec. 2 Quantization of Free Field Theory 1. The Lagrangian form of the classical field theory can be written in Hamiltonian form so that it’s easier to be quantized: 1 1 1 2 H d 3 x 2 m 2 2 2 2 2 2. As explained, field theory is actually a collection of SHOs. Simple harmonic oscillator can be quantized using the raising and lowering operators a and a†: (briefly reviewed on p20 after Eq. (2.22), and certainly explained in your quantum mechanics textbook) a, a 1 1 H aa 2, n a n 0 3. The scalar field theory could be quantized like in quantum mechanics. It means the fields become operators. For operators, we need to specify their commutation relation. Using Canonical Quantization procedure as in QM: ( x), y i (3) ( x y) 4. The scalar field when Fourier transformed constitutes a series of SHO. And the quantization of scalar field as specified in 3. is equivalent to quantizing the SHO’s: d3p (x) (2 ) 3 a H p 1 2 p a p e ip x a p e ip x 3 , a p ' 2 (3) p p' d3p 2 3 p a p a p 1 a p , a p ' 2 5. The energy eigenstate states can be constructed like in SHO and the energy eigenvalues look exactly like particle states with various particle numbers. This is also true for momentum. See the paragragh after Eq. (2.33) on P. 9. So field, when quantized, become particles. Even better, these are relativistic particles. 8 For field, the space coordinates x correspond to the indices to specify degrees of freedom. So the above commutation relation can be easily generalized to the field operator ϕ(x) and its conjugate momentum π(x): ( x, t ), y, t i (3) ( x y), ( x, t ), y, t 0, ( x, t ), y, t 0 These relations are equivalent to corresponding commutation relations between the a operators. It can be shown that the above relation implies: 3 a p , a p ' 2 3 p p' and vise versa: This is exactly commutation relation of the a operators for a series of quantum simple 9 harmonic oscillator (SHO). For a quantum SHO: The spectrum of SHO is like a ladder. The operator a+ can be used to raise the energy by one quantum while the operator a can be used to lower the energy by one quantum: The operator a+ is called Raising Operator while the operator a Lowering Operator. It has been shown above that Field theory consists of a series of SHO. Hence Quantum Field Theory is just a series of quantum SHO. Now the Hamiltonian of QFT can be written in terms of a’s: 10 d3p (x) (2 ) 3 a n p 1 2 p a p e ip x a p e ip x 0 p, n P p, n np p, n H p, n nE p p, n 11 12 Dirac Field 13 14