Quantum systems governed by non-Hermitian Hamiltonians with $\PT$ symmetry are special in having real energy eigenvalues bounded below and unitary time evolution. We argue that $\PT$ symmetry may also be important and present at the level of Hermitian quantum field theories because of the process of renormalisation. In some quantum field theories renormalisation leads to $\PT$-symmetric effective Lagrangians. We show how $\PT$ symmetry may allow interpretations that evade ghosts and instabilities present in an interpretation of the theory within a Hermitian framework. From the study of examples $\PT$-symmetric interpretation is naturally built into a path integral formulation of quantum field theory; there is no requirement to calculate explicitly the $\PT$ norm that occurs in Hamiltonian quantum theory. We discuss examples where $\PT$-symmetric field theories emerge from Hermitian field theories due to effects of renormalization. We also consider the effects of renormalization on field theories that are non-Hermitian but $\PT$-symmetric from the start.
Center for Theoretical Physics and College of Physics, Jilin University（吉林大学理论物理中心与物理学院）；Institut für Theoretische Physik, Universität Heidelberg（海德堡大学理论物理研究所）；ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung mbH（GSI重离子研究中心极端物质研究所EMMI）；Department of Physics and Astronomy, Kwansei Gakuin University（关西学院大学物理天文学系）
B.R.S. renormalisation of some on-shell closed algebras of symmetry transformations : the example of supersymmetric non-linear sigma models : the N=1 case
We compute Passarino-Veltman (PV) reduction for tensor loop integrals, that appear in open field theories. We apply these results to open-Yukawa theory and compute the self-energy correction of the fields. We found that non-local divergences show up in the one loop correction to the fermionic self-energy. These non-local divergences do not disappear even if the tree level theory is chosen to satisfy the trace preserving condition of the density matrix.
There exist renormalisation schemes that explicitly preserve the scale invariance of a theory at the quantum level. Imposing a scale invariant renormalisation breaks renormalisability and induces new non-trivial operators in the theory. In this work, we study the effects of such scale invariant renormalisation procedures. On the one hand, an explicitly quantum scale invariant theory can emerge from the scale invariant renormalisation of a scale invariant Lagrangian. On the other hand, we show how a quantum scale invariant theory can equally emerge from a Lagrangian visibly breaking scale invar…
In physics we attempt to infer the rules governing a system given only the results of imprecise measurements. This is an ill-posed problem because certain features of the system's state cannot be resolved by the measurements. However, by ignoring the irrelevant features, an effective theory can be made for the remaining observable relevant features. We explain how these relevant and irrelevant degrees of freedom can be concretely characterised using quantum distinguishability metrics, thus solving the ill-posed inference problem. This framework then allows us to provide an information-theoreti…
The canonical approach to quantum gravity has been put on a firm mathematical foundation in the recent decades. Even the quantum dynamics can be rigorously defined, however, due to the tremendously non-polynomial character of the gravitational interaction, the corresponding Wheeler-DeWitt operator valued distribution suffers from quantisation ambiguities that need to be fixed. In a very recent series of works we have employed methods from constructive quantum field theory in order to address those ambiguities. Constructive QFT trades quantum fields for random variables and measures thereby phr…
We propose a general formulation of the renormalisation group as a family of quantum channels which connect the microscopic physical world to the observable world at some scale. By endowing the set of quantum states with an operationally motivated information geometry, we induce the space of Hamiltonians with a corresponding metric geometry. The resulting structure allows one to quantify information loss along RG flows in terms of the distinguishability of thermal states. In particular, we introduce a family of functions, expressible in terms of two-point correlation functions, which are non i…
Rank-d Tensorial Group Field Theories are quantum field theories defined on a group manifold $G^{\times d}$, which represent a non-local generalization of standard QFT, and a candidate formalism for quantum gravity, since, when endowed with appropriate data, they can be interpreted as defining a field theoretic description of the fundamental building blocks of quantum spacetime. Their renormalisation analysis is crucial both for establishing their consistency as quantum field theories, and for studying the emergence of continuum spacetime and geometry from them. In this paper, we study the ren…
In physics one attempts to infer the rules governing a system given only the results of imperfect measurements. Hence, microscopic theories may be effectively indistinguishable experimentally. We develop an operationally motivated procedure to identify the corresponding equivalence classes of states, and argue that the renormalisation group arises from the inherent ambiguities associated with the classes: one encounters flow parameters as, e.g., a regulator, a scale, or a measure of precision, which specify representatives in a given equivalence class. This provides a unifying framework and id…
That the exact quantum S-matrix of $\text{T}\bar{\text{T}}$-deformed field theories is known has interesting consequences for their perturbative renormalisation. Recent investigations into the interplay between renormalisation and integrable deformations have focused on those cases where the (undeformed) seed theories are integrable. In this paper, we study the perturbative renormalisation of non-integrable $\text{T}\bar{\text{T}}$-deformed field theories, focusing on a theory of scalars with a quartic interaction. We show that in this theory, the 4-point 1-loop S-matrix exhibits features simi…
For arbitrary four-dimensional quantum field theories with scalars and fermions, renormalisation group equations in the $\overline{\text{MS}}$ scheme are investigated at three-loop order in perturbation theory. Collecting literature results, general expressions are obtained for field anomalous dimensions, Yukawa interactions, as well as fermion masses. The renormalisation group evolution of scalar quartic, cubic and mass terms is determined up to a few unknown coefficients. The combined results are applied to compute the renormalisation group evolution of the gaugeless Litim-Sannino model.
We use the Wilsonian functional Renormalisation Group (RG) to study quantum corrections for the Higgs inflationary action including the effect of gravitons, and analyse the leading-order quantum gravitational corrections to the Higgs' quartic coupling, as well as its non-minimal coupling to gravity and Newton's constant, at the inflationary regime and beyond. We explain how within this framework the effect of Higgs and graviton loops can be sufficiently suppressed during inflation, and we also place a bound on the corresponding value of the infrared RG cut-off scale during inflation. Finally, …
These are introductory lecture notes aimed at beginning graduate students covering fundamental concepts and ideas behind the renormalisation group. Our main goal is to motivate it and then explore its consequences, in the context of quantum field theory. We shall reproduce the standard results without needing any mention of cancellation of infinities. To keep the story simple and compact, we'll ignore the role of symmetry in this game. We'll also restrict to scalar fields, and not mention neither fermions nor gauge fields. Moreover, we'll only briefly mention the connection with statistical fi…
The scaling behaviour of euclidean quantum gravity at an asymptotically safe critical point is studied by means of the exact renormalisation group. Gauge independence is ensured via a specific parameterisation of metric fluctuations introduced in a recent paper. Within a non-perturbative approximation the beta function for Newton's constant takes a simple form to all orders in $\hbar$. A UV fixed point is found to exist for $d\leq 7$ spacetime dimensions at which the critical scaling can be assessed. The critical exponent for the Newton's constant $ν$ is found to be regulator independent close…
Non-differentiable potentials, such as the V-shaped (linear) potential, appear in various areas of physics. For example, the effective action for branons in the framework of the brane world scenario contains a Liouville-type interaction, i.e., an exponential of the V-shaped function. Another example is coming from particle physics when the standard model Higgs potential is replaced by a periodic self-interaction of an N-component scalar field which depends on the length, thus it is O(N) symmetric. We first compare classical and quantum dynamics near non-analytic points and discuss in this cont…
Hamiltonian Renormalisation, as defined within this series of works, was derived from covariant Wilson renormalisation via Osterwalder-Schrader reconstruction. As such it directly applies to QFT with a true (physical) Hamiltonian bounded from below. The validity of the scheme was positively tested for free QFT in any dimension with or without Abelian gauge symmetries of Yang-Mills type. The aim of this Hamiltonian renormalisation scheme is to remove quantisation ambiguities of Hamiltonians in interacting QFT that remain even after UV and IR regulators are removed as it happens in highly non-li…
We derive new functional renormalisation group flows for quantum gravity, in any dimension. The key new achievement is that the equations apply for any theory of gravity whose underlying Lagrangian $\sim f(R_{μνρσ})$ is a function of the Riemann tensor and the inverse metric. The results centrally exploit the benefits of maximally symmetric spaces for the evaluation of operator traces. The framework is highly versatile and offers a wide range of new applications to study quantum gravitational effects in extensions of Einstein gravity, many of which have hitherto been out of reach. The phase di…
We study quantum corrections to Friedmann-Robertson-Walker cosmology with a scalar field under the assumption that the dynamics are subject to renormalisation group improvement. We use the Bianchi identity to relate the renormalisation group scale to the scale factor and obtain the improved cosmological evolution equations. We study the solutions of these equations in the renormalisation group fixed point regime, obtaining the time-dependence of the scalar field strength and the Hubble parameter in specific models with monomial and trinomial quartic scalar field potentials. We find that power-…
We compute the two-loop $β$-function of scalar and spinorial quantum electrodynamics as well as pure Yang-Mills and quantum chromodynamics using the background field method in a fully quadridimensional setup using Implicit Regularization (IREG). Moreover, a thorough comparison with dimensional approaches such as conventional dimensional regularization (CDR) and dimensional reduction (DRED) is presented. Subtleties related to Lorentz algebra contractions/symmetric integrations inside divergent integrals as well as renormalisation schemes are carefully discussed within IREG where the renormalisa…
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Radiative symmetry breaking (RSB) is a theoretically appealing framework for the generation of mass scales through quantum effects. It can be successfully implemented in models with extended scalar and gauge sectors. We provide a systematic analysis of RSB in such models: we review the common approximative methods of studying RSB, emphasising their limits of applicability and discuss the relevance of the relative magnitudes of tree-level and loop contributions as well as the dependence of the results on the renormalisation scale. The general considerations are exemplified within the context of…
Worldline holography states that within the framework of the worldline approach to quantum field theory, sources of a quantum field theory over Mink$_4$ naturally form a field theory over AdS$_5$ {\sl to all orders} in the elementary fields and in the sources of arbitrary spin. (Such correspondences are also available for other pairs of spacetimes, not only Mink$_4\leftrightarrow\mathrm{AdS}_5$.) Schwinger's proper time of the worldline formalism is automatically grouped with the physical four spacetime dimensions into an AdS$_5$ geometry. We show that the worldline holographic effective actio…
We study, as a model of Lorentz symmetry breaking, the quantisation and renormalisation of an extension of QED in a flat spacetime where the photons and electrons propagate differently and do not share the same lightcone. We will refer to this model as Bimetric QED (BIMQED). As a preliminary we discuss the formulation of electrodynamics in a pre-metric formalism showing nevertheless that there is, on the basis of a simple criteron, a preferred metric. Arising from this choice of metric is a Weyl-like tensor (WLT). The Petrov classification of the WLT gives rise to a corresponding classificatio…
We introduce a novel momentum space entanglement renormalization group (MERG) scheme for the topologically ordered (T.O.) ground state of the 2D Hubbard model on a square lattice (\cite{anirbanmotti,anirbanmott2}) using a unitary quantum circuit comprised of non-local unitary gates. At each MERG step, the unitary quantum circuit disentangles a set of electronic states, thereby transforming the tensor network representation of the many-particle state. By representing the non-local unitary gate as a product of two-qubit disentangler gates, we provide an entanglement holographic mapping (EHM) rep…
By means of simple models in a flat spacetime manifold we examine some of the issues that arise when quantizing interacting quantum fields in multi-metric backgrounds. In particular we investigate the maintenance of a causal structure in the models. In this context we introduce and explain the relevance of an interpolating metric that is a superposition of the individual metrics in the models. We study the renormalisation of a model with quartic interactions and elucidate the structure of the renormalisation group and its implications for Lorentz symmetry breakdown.
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In a recent proposal we applied methods from constructive QFT to derive a Hamiltonian Renormalisation Group in order to employ it ultimately for canonical quantum gravity. The proposal was successfully tested for free scalar fields and thus a natural next step is to test it for free gauge theories. This can be done in the framework of reduced phase space quantisation which allows using techniques developed earlier for scalar field theories. In addition, in canonical quantum gravity one works in representations that support holonomy operators which are ill defined in the Fock representation of …
We present an analytic computation of an explicit renormalisation group flow for cosmological states in loop quantum gravity. A key ingredient in our analysis are Perelomov coherent states for the Lie group SU(1,1) whose representation spaces are embedded into the standard loop quantum cosmology (LQC) Hilbert space. The SU(1,1) group structure enters our analysis by considering a classical set of phase space functions that generates the Lie algebra su(1,1). We implement this Poisson algebra as operators on the LQC Hilbert space in a non-anomalous way. This task requires a rather involved order…
