Quantum Mechanics for Mathematicians
causality once interactions are allowed, with influence from what happens at one point in space-time traveling to another at fas ...
(see equation 43.7) are related by this non-standard conjugation (only on real solutions isα(p) the complex conjugate ofα(p)). A ...
Digression. Quantum field theory textbooks often contain a discussion of a non-positive Hermitian inner product on the space of ...
Here the two different integrandsH(x) are related (as in the non-relativistic case) by integration by parts, so these just diffe ...
A(α(p))∈M−Jr=H 1 →a(α(p)) HereA(α(p)) is the positive energy solution of the (complexified) Klein-Gordon equation with initial d ...
To make physical sense of the quanta in the relativistic theory, assigning all non-vacuum states a positive energy, we take such ...
∂ ∂t Π = [̂ Π̂,−iĤ] = (∆−m^2 )̂Φ which have as solution the following equation for the time-dependent field op- erator: Φ(̂t,x) ...
As in the non-relativistic case (see section 12.5), this is a distribution that can be defined as a boundary value of an analyti ...
In the non-relativistic case the Green’s functionĜonly had one pole, atp 0 = |p|^2 / 2 m. Two possible choices of how to extend ...
For a fixed ultraviolet cutoff, corresponding physically to only allowing fields with momentum components smaller than a given v ...
Chapter 44 Symmetries and Relativistic Scalar Quantum Fields Just as for non-relativistic quantum fields, the theory of free rel ...
simplest case of fields taking values inR^2 orC, one gets a theory of charged relativistic particles, with antiparticles now dis ...
and integrate the analog ofμLoverR^3 to get an appropriate moment map for the field theory case. This gives a quadratic function ...
Note that the vacuum state| 0 〉is an eigenvector forQ̂andĤwith both eigen- values 0: it has zero energy and zero charge. States ...
whereH+ 1 will be positive energy solutions withJC = +i andH− 1 will be positive energy solutions withJC=−i. Taking as beforeα 1 ...
Using these creation and annihilation operators, we can define position space field operators analogous to the ones given by equ ...
It will also give a representation ofU(1) on states, with the state space de- composing into sectors each labeled by the integer ...
Quantization should give unitary operatorsU(a,Λ), which act on field operators by Φ̂→U(a,Λ)Φ(̂x)U(a,Λ)−^1 =Φ(Λ̂ x+a) (44.6) TheU ...
44.2.1 Translations For time translations, we have already found the Hamiltonian operatorĤ, which gives the infinitesimal trans ...
44.2.2 Rotations We can use the same method as for translations to find the quadratic combi- nations of coordinates onMcorrespon ...
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