Quantum Field Theory I

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1.3. RELATIVITY AND QUANTUM THEORY 39 1.3 Relativity and <strong>Quantum</strong> <strong>Theory</strong> Actually, as to the quantum fields, the keyword is relativity. Even if QFT is useful also in the nonrelativistic context (see the previous section), the very notion of the quantum field originated from an endeavor to fit together relativity and quantum theory. This is a nontrivial task: to formulate a relativistic quantum theory is significantly more complicated than it is in a nonrelativistic case. The reason is that specification of a Hamiltonian, the crucial operator of any quantum theory, is much more restricted in relativistic theories. To understand the source of difficulties, it is sufficient to realize that to have a relativistic quantum theory means to have a quantum theory with measurable predictions which remain unchanged by the relativistic transformations. The relativistic transformations at the classical level are the space-time transformationsconservingthe intervalds = η µν dx µ dx ν , i.e. boosts, rotations,translations and space-time inversions (the list is exhaustive). They constitute a group. Now to the quantum level: whenever macroscopic measuring and/or preparing devices enjoy a relativistic transformation, the Hilbert (Fock) space should transformaccordingtoacorrespondinglinear 42 transformation. Itisalsoalmost self-evident that the quantum transformation corresponding to a composition of classical transformations should be equal to the composition of quantum transformations corresponding to the individual classical transformations. So at the quantumleveltherelativistictransformationsshouldconstitutearepresentation of the group of classical transformations. The point now is that the Hamiltonian, as the time-translations generator, is among the generators of the Poincaré group (boosts, rotations, translations). Consequently, unlike in a nonrelativistic QM, in a relativistic quantum theory one cannot specify the Hamiltonian alone, one has rather to specify it within a complete representation of the Poincaré algebra. This is the starting point of any effort to get a relativistic quantum theory, even if it is not always stated explicitly. The outcome of such efforts are quantum fields. Depending on the philosophy adopted, they use to emerge in at least two different ways. We will call them particle-focused and field-focused. The particle-focused approach, represented mostly by the Weinberg’s book, is very much in the spirit of the previous section. One starts from the Fock space, which is systematically built up from the 1-particle Hilbert space. Creationandannihilationoperatorsarethendefinedasverynaturalobjects,namely as maps from the n-particle subspace into (n±1)-particle ones, and only afterwards quantum fields are built from these operators in a bit sophisticated way (keywords being cluster decomposition principle and relativistic invariance). The field-focused approach (represented by Peskin–Schroeder, and shared by the majority of textbooks on the subject) starts from the quantization of a classical field, introducing the creation and annihilation operators in this way as quantum incarnations of the normal mode expansion coefficients, and finally providingFockspaceasthe worldforthese operatorstolivein. The logicbehind this formal development is nothing else but construction of the Poincaré group generators. So, again, the corner-stone is the relativistic invariance. 42 Linearity of transformations at the quantum level is necessary to preserve the superposition principle. The symmetry transformations should not change measurable things, which would not be the case if the superposition |ψ〉 = c 1 |ψ 1 〉 + c 2 |ψ 2 〉 would transform to T |ψ〉 ≠ c 1 T |ψ 1 〉+c 2 T |ψ 2 〉.
40 CHAPTER 1. INTRODUCTIONS 1.3.1 Lorentz and Poincaré groups This is by no means a systematic exposition to the Lorentz and Poincarégroups and their representations. It is rather a summary of important relations, some of which should be familiar (at some level of rigor) from the previous courses. the groups The classical relativistic transformations constitute a group, the corresponding transformations at the quantum level constitute a representation of this group. The (active) group transformations are x µ → Λ µ ν xν +a µ where Λ µ ν are combined rotations, boosts and space-time inversions, while a µ describe translations. The rotations around (and the boosts along) the space axes are R 1 (ϑ) = R 2 (ϑ) = R 3 (ϑ) = ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ 1 0 0 0 0 1 0 0 0 0 cosϑ −sinϑ 0 0 sinϑ cosϑ 1 0 0 0 0 cosϑ 0 sinϑ 0 0 1 0 0 −sinϑ 0 cosϑ 1 0 0 0 0 cosϑ −sinϑ 0 0 sinϑ cosϑ 0 0 0 0 1 ⎞ ⎟ ⎠ B 1 (β) = ⎞ ⎟ ⎠ B 2 (β) = ⎞ ⎟ ⎠ B 3 (β) = ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ chβ −shβ 0 0 −shβ chβ 0 0 0 0 1 0 0 0 0 1 chβ 0 −shβ 0 0 1 0 0 −shβ 0 chβ 0 0 0 0 1 chβ 0 0 −shβ 0 1 0 0 0 0 1 0 −shβ 0 0 chβ where ϑ is the rotation angle and tanhβ = v/c. They constitute the Lorentz group. It is a non-compact (because of β ∈ (−∞,∞)) Lie group. The translations along the space-time axes are T 0 (α) = ⎛ ⎜ ⎝ α 0 0 0 ⎞ ⎟ ⎠ T 1(α) = ⎛ ⎜ ⎝ 0 α 0 0 ⎞ ⎟ ⎠ T 2(α) = ⎛ ⎜ ⎝ 0 0 α 0 ⎞ ⎟ ⎠ T 3(α) = Together with the boosts and rotations they constitute the Poincaré group. It is a non-compact (on top of the non-compactness of the Lorentz subgroup one has α ∈ (−∞,∞)) Lie group. The space-time inversions are four different diagonal matrices. The time inversion is given by I 0 = diag(−1,1,1,1), the three space inversions are given by I 1 = diag(1,−1,1,1) I 2 = diag(1,1,−1,1) I 3 = diag(1,1,1,−1). ⎛ ⎜ ⎝ 0 0 0 α ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ ⎞ ⎟ ⎠
Page 1: Quantum Field Theory I Martin Mojž
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