New PDF release: A story of light: a short introduction to quantum field

By M. Y. Han

ISBN-10: 9812560343

ISBN-13: 9789812560346

Offers the fundamental points of relativistic quantum box idea with minimum use of arithmetic. It covers the advance of quantum box conception from the unique quantization of electromagnetic box to the gauge box thought of interactions between quarks and leptons.

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Additional resources for A story of light: a short introduction to quantum field theory of quarks and leptons

Example text

The canonical quantization procedure in terms of the commutators, as shown above, is rooted in the Poisson bracket formalism of Hamilton’s formulation of mechanics, as discussed in Chapter 3. It leads to a successful theory of quantized fields for the Klein–Gordon and electromagnetic fields, that is, those that represent particles of spin zero and one, in fact, of all integer values of spin. The particles of half-integer spins, half, and one and half, and so on, must satisfy the Pauli exclusion principle and the fields that represent these particles, the Dirac field in particular, must be quantized not by commutators [A, B] ≡ AB − BA but by anticommutators {A, B} ≡ AB + BA.

Introducing a new notation ωk ≡ + k2 + m2 with only the + sign, either k0 = +ωk or k0 = −ωk . 2 This decomposition, which is basic to all relativistic fields, matter fields as well as the electromagnetic field, has nothing to do with field quantization and is rooted in the quadratic nature of Einstein’s energy–momentum formula. The plane-wave solutions can be written in the form: d3 k(a(k)fk (x) + a∗ (k)fk∗ (x)) φ(x) = where fk (x) = 1 (2π)3 2ω e−ikx k and fk∗ (x) = 1 (2π)3 2ω e+ikx . k The integral is over d3 k only and a(k) and a∗ (k) are the respective Fourier transforms for “positive frequency” and “negative frequency” parts.

The Dirac equation imposes further conditions over and beyond the Klein–Gordon equation — very stringent interrelations among the components and their first derivatives — among the four components of the solution. This can be seen when the Dirac equation is fully written out in 4 × 4 matrix format using an explicit representation of γ-matrices such as shown above. The Dirac equation is the centerpiece of relativistic quantum mechanics. All textbooks on the subject devote a substantial amount of the contents to detailing all aspects of this equation — proof of its relativistic covariance, the algebraic properties of Dirac matrices, as γ matrices are called, the bilinear covariants built from its fourcomponent solutions, and many others — and, in fact, virtually all textbooks on quantum field theory also include extensive discussions about the equation, before embarking on the subject of field quantization.

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A story of light: a short introduction to quantum field theory of quarks and leptons by M. Y. Han


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