Renormalization Group Invariant Objects of a Quantum Field Theory
Renormalization group invariant objects of a quantum field theorybeginequation beginsplit fracd(g_bmu^epsilon)dlogmu^2&=fracmu2fracd(g_bmu^epsilon)dmu &=fracmu2left[mu^epsilonfracdg_bdmug_bfracdmu^epsilondmuight] endsplit endequation By definition, the bare coupling does not depend on the renormalization scale $mu$. Hence beginequation fracd(g_bmu^epsilon)dlogmu^2=fracepsilon g_b2mu^epsilon, endequation which vanishes as $epsilonightarrow 0$.Edit: Notice that the authors define beginequation a_B=Z_as a_s, endequation where $a_B$ is the bare coupling and $a_s$ is the renormalized coupling, with beginequation a_sequivfracg(mu^2)16pi^2. endequation In order to keep the coupling $g$ dimensionless in dimensional regularization, we must introduce the dimensionfull quantity $mu$, so that in $d=4-2epsilon$ dimensions we have beginequation gightarrow mu^epsilong, endequation or beginequation a_sightarrow mu^2epsilona_s. endequation Hence beginequation a_Bmu^2epsilon=Z_asa_smu^2epsilon. endequation Ordinarily (in my experience) we conclude that the bare coupling itself is invariant under the renormalization group flow because we have already included the scale $mu$ in its definition, i.e. beginequation a_B=Z_asa_smu^2epsilon. endequation However, based on the author's convention, we must include the scale $mu$ on both sides of this equation. Now that we have ensured that the dimensions will be preserved, we can say that beginequation fracd(a_Bmu^2epsilon)dlogmu^2=0. endequationâ â â â â âMomentum eigenstates in an interacting quantum field theoryI am thinking about this too. I decided on the following. The one particle states we are talking about are always one particle states of the Full interacting theory. Therefore, the in field operators are not operators of the free theory. They are just the field operators at t -> inf. They are still very much operators of the full theory, acting on states in the Hilbert space of the full theory. However, the infield operators have the property of satisfying the Klein-Gordon equation. So actually the single particle states are created by them. When you hit a single particle state with an infield operator you just create or destroy 1 particle. Bear in mind this is all happening in the full interaction theory Hilbert space. We are not going back and forth between free and interacting theory!What happens when you use a field operator (not infield, just plain old phi) to act on single particle states? You get a mess, the state becomes a superposition of many particles. Even though that happens, eqn 16. 36 still becomes zero, because it just relies on the fact that the single particle states are momentum states on the mass shell (p2 - m2 = 0), and some properties of the P operator (of the full theory).So in summary, all the states, operators and fields (including infields) are of the Full theory. There is no contradiction!One thing Im struggling to see is why in the interacting theory, you need to rescale the fields. Its easy to get lost in the long chain of arguments. Is there an example or analogy from perturbation theory of single particle QM?â â â â â âQuantum field theoryQuantization of the electromagnetic fieldIn 1910, Peter Debye derived Planck's law of black-body radiation from a relatively simple assumption. He decomposed the electromagnetic field in a cavity into its Fourier modes, and assumed that the energy in any mode was an integer multiple of h displaystyle hu , where displaystyle u is the frequency of the electromagnetic mode. Planck's law of black-body radiation follows immediately as a geometric sum. However, Debye's approach failed to give the correct formula for the energy fluctuations of black-body radiation, which were derived by Einstein in 1909. In 1925, Born, Heisenberg and Jordan reinterpreted Debye's concept in a key way. As may be shown classically, the Fourier modes of the electromagnetic field-a complete set of electromagnetic plane waves indexed by their wave vector k and polarization state-are equivalent to a set of uncoupled simple harmonic oscillators. Treated quantum mechanically, the energy levels of such oscillators are known to be E = n h displaystyle E=nhu , where displaystyle u is the oscillator frequency. The key new step was to identify an electromagnetic mode with energy E = n h displaystyle E=nhu as a state with n displaystyle n photons, each of energy h displaystyle hu . This approach gives the correct energy fluctuation formula. Dirac took this one step further. He treated the interaction between a charge and an electromagnetic field as a small perturbation that induces transitions in the photon states, changing the numbers of photons in the modes, while conserving energy and momentum overall. Dirac was able to derive Einstein's A i j displaystyle A_ij and B i j displaystyle B_ij coefficients from first principles, and showed that the Bose-Einstein statistics of photons is a natural consequence of quantizing the electromagnetic field correctly (Bose's reasoning went in the opposite direction; he derived Planck's law of black-body radiation by assuming B-E statistics). In Dirac's time, it was not yet known that all bosons, including photons, must obey Bose-Einstein statistics. Dirac's second-order perturbation theory can involve virtual photons, transient intermediate states of the electromagnetic field; the static electric and magnetic interactions are mediated by such virtual photons. In such quantum field theories, the probability amplitude of observable events is calculated by summing over all possible intermediate steps, even ones that are unphysical; hence, virtual photons are not constrained to satisfy E = p c displaystyle E=pc , and may have extra polarization states; depending on the gauge used, virtual photons may have three or four polarization states, instead of the two states of real photons. Although these transient virtual photons can never be observed, they contribute measurably to the probabilities of observable events. Indeed, such second-order and higher-order perturbation calculations can give apparently infinite contributions to the sum. Such unphysical results are corrected for using the technique of renormalization. Other virtual particles may contribute to the summation as well; for example, two photons may interact indirectly through virtual electron-positron pairs. Such photon-photon scattering (see two-photon physics), as well as electron-photon scattering, is meant to be one of the modes of operations of the planned particle accelerator, the International Linear Collider. In modern physics notation, the quantum state of the electromagnetic field is written as a Fock state, a tensor product of the states for each electromagnetic mode | n k 0 | n k 1 | n k n ... displaystyle |n_k_0angle otimes |n_k_1angle otimes dots otimes |n_k_nangle dots where | n k i displaystyle |n_k_iangle represents the state in which n k i displaystyle ,n_k_i photons are in the mode k i displaystyle k_i . In this notation, the creation of a new photon in mode k i displaystyle k_i (e.g., emitted from an atomic transition) is written as | n k i | n k i 1 displaystyle |n_k_iangle ightarrow |n_k_i1angle . This notation merely expresses the concept of Born, Heisenberg and Jordan described above, and does not add any physics. As a gauge bosonThe electromagnetic field can be understood as a gauge field, i.e., as a field that results from requiring that a gauge symmetry holds independently at every position in spacetime. For the electromagnetic field, this gauge symmetry is the Abelian U(1) symmetry of complex numbers of absolute value 1, which reflects the ability to vary the phase of a complex field without affecting observables or real valued functions made from it, such as the energy or the Lagrangian. The quanta of an Abelian gauge field must be massless, uncharged bosons, as long as the symmetry is not broken; hence, the photon is predicted to be massless, and to have zero electric charge and integer spin. The particular form of the electromagnetic interaction specifies that the photon must have spin 1; thus, its helicity must be displaystyle pm hbar . These two spin components correspond to the classical concepts of right-handed and left-handed circularly polarized light. However, the transient virtual photons of quantum electrodynamics may also adopt unphysical polarization states. In the prevailing Standard Model of physics, the photon is one of four gauge bosons in the electroweak interaction; the other three are denoted W, W and Z0 and are responsible for the weak interaction. Unlike the photon, these gauge bosons have mass, owing to a mechanism that breaks their SU(2) gauge symmetry. The unification of the photon with W and Z gauge bosons in the electroweak interaction was accomplished by Sheldon Glashow, Abdus Salam and Steven Weinberg, for which they were awarded the 1979 Nobel Prize in physics. Physicists continue to hypothesize grand unified theories that connect these four gauge bosons with the eight gluon gauge bosons of quantum chromodynamics; however, key predictions of these theories, such as proton decay, have not been observed experimentally. Hadronic propertiesMeasurements of the interaction between energetic photons and hadrons show that the interaction is much more intense than expected by the interaction of merely photons with the hadron's electric charge. Furthermore, the interaction of energetic photons with protons is similar to the interaction of photons with neutrons in spite of the fact that the electric charge structures of protons and neutrons are substantially different. A theory called Vector Meson Dominance (VMD) was developed to explain this effect. According to VMD, the photon is a superposition of the pure electromagnetic photon which interacts only with electric charges and vector mesons. However, if experimentally probed at very short distances, the intrinsic structure of the photon is recognized as a flux of quark and gluon components, quasi-free according to asymptotic freedom in QCD and described by the photon structure function. A comprehensive comparison of data with theoretical predictions was presented in a review in 2000. Contributions to the mass of a systemThe energy of a system that emits a photon is decreased by the energy E displaystyle E of the photon as measured in the rest frame of the emitting system, which may result in a reduction in mass in the amount E / c 2 displaystyle E/c^2 . Similarly, the mass of a system that absorbs a photon is increased by a corresponding amount. As an application, the energy balance of nuclear reactions involving photons is commonly written in terms of the masses of the nuclei involved, and terms of the form E / c 2 displaystyle E/c^2 for the gamma photons (and for other relevant energies, such as the recoil energy of nuclei). This concept is applied in key predictions of quantum electrodynamics (QED, see above). In that theory, the mass of electrons (or, more generally, leptons) is modified by including the mass contributions of virtual photons, in a technique known as renormalization. Such "radiative corrections" contribute to a number of predictions of QED, such as the magnetic dipole moment of leptons, the Lamb shift, and the hyperfine structure of bound lepton pairs, such as muonium and positronium. Since photons contribute to the stress-energy tensor, they exert a gravitational attraction on other objects, according to the theory of general relativity. Conversely, photons are themselves affected by gravity; their normally straight trajectories may be bent by warped spacetime, as in gravitational lensing, and their frequencies may be lowered by moving to a higher gravitational potential, as in the Pound-Rebka experiment. However, these effects are not specific to photons; exactly the same effects would be predicted for classical electromagnetic waves.