From 72e37a192ad99e9784e0c226b150994f030b490e Mon Sep 17 00:00:00 2001 From: "Tobias J. Osborne" Date: Tue, 14 Apr 2015 10:41:11 +0200 Subject: [PATCH] Minor corrections --- What is a quantum field state.pdf | Bin 1776203 -> 1776461 bytes What is a quantum field state.tex | 74 +++++++++++++++--------------- What-is-a-quantum-field-state.bib | 7 +++ 3 files changed, 43 insertions(+), 38 deletions(-) diff --git a/What is a quantum field state.pdf b/What is a quantum field state.pdf index f08fa46d0e3ab7ca2428af09f66cb2d3ecb75406..70c2c31dfedb9cbfed10d9968d51e5cb8462ef10 100644 GIT binary patch delta 97951 zcmV)JK)b)ov~10`Y>*@ZGBKB-+ZiZ-g;_~++%^`z>sNHmWvCj9<^pcHOle{(b|#s` zrB3Of=zu1nZpK{N1Y5Fd{`|ZRY*MCcE)C$}0eE=p2X?U8Z-UKF7r}She{C<`-Xv-h z27VC~;b!}|36sK)l1*9^eh?O$ZMC`g?mk>!7OA(ryi!H(eZ0lC@UD6I>5uDw|7`#G zm1#F0-~aNC>9<>4!wcX4^YUT)^K>kn8EtaENK=JJUL~PlB;n>N%KaeYBd>RM9Q0)r zdFAN8S{#M5#jrL|1k@jyyvbgo!VgSD|-;Z8S5`6=2@hEgp7iG>ZC$ZW6q@18p8E@4C8wef#6h76%EO zGz^Ah0xfox=}LpbMuFT8SVlOG*CyVs(Ip=AnTLnIkxfO1q9`CVy|v7A)Pt^iT=mrl 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z1}qGif#`u}SI_~2!C?>T;ZgrD8JYl@$9tpvf79t7Ob0S3$A@D9wSbV(JsI?Q5ga_1 z19##5P6HY0AI9}R{s(9P;Ia_Ky=RP&P-DSm8(JSg2Btn_Ew~;-o{E6+z6Ujb>H+?@ z#~1dK!oOr-&p#FSWPsQ~!VhAPh2lJrfzpJo#pC|WyQ{N_wcSl;YH?x}3#*6;UPcN$ zcmk6JAtxs-D~kqv6eWX_LdzkfrEyr9%aWYp|6i5soRX5%F0LlduI|n^EvNxCf`Oi) L=Hru9lcWAWM2TZ3 diff --git a/What is a quantum field state.tex b/What is a quantum field state.tex index 89909d7..6211f52 100644 --- a/What is a quantum field state.tex +++ b/What is a quantum field state.tex @@ -49,47 +49,47 @@ \begin{document} \begin{abstract} -There has recently been in a fruitful interplay of ideas between quantum information theory and high energy physics, especially in the context of quantum simulation, the AdS/CFT correspondence, and the black hole information loss paradox. However, a core difficulty faced by quantum information theorists -- who are usually concerned with qubits -- interested in these emergent and vibrant areas is the necessity of dealing with quantum fields. The primary purpose of these notes is to lower the entry barrier for quantum information theorists to work on such exciting and challenging topics by explaining what, in a quantum information friendly way, a quantum field state actually is. We describe the Wilsonian formulation of quantum field theory as an effective theory and explain how this leads naturally to a definition, independent of lagrangians, of quantum field states which is better adapted to, e.g., tensor networks. We hope that there is something here for high-energy theorists as well, if only to see how someone from ``the other side'' thinks about complex quantum systems. +There has been in a fruitful interplay of ideas between quantum information theory and high energy physics, especially in the context of quantum simulation, the AdS/CFT correspondence, and the black hole information loss paradox. However, a core difficulty faced by quantum information theorists -- who are usually concerned with qubits -- interested in these emergent and vibrant areas is the necessity of dealing with quantum fields. The primary purpose of these notes is to lower the entry barrier for quantum information theorists to work on these topics by explaining what, in a quantum information friendly way, a quantum field state actually is. We describe the Wilsonian formulation of quantum field theory as an effective theory and explain how this leads naturally to a definition, independent of lagrangians, of quantum field states which is better adapted to, e.g., tensor networks. We hope that there is something here for high-energy theorists as well, if only to see how someone from ``the other side'' thinks about complex quantum systems. \end{abstract} \maketitle \section{Introduction} -Quantum field theory (QFT) has become, thanks in no small part to Wilson -\cite{wilson_renormalization_1974,wilson_renormalization_1975}, an immensely powerful calculational machine to solve and approximate a wide variety of physical problems from the fundamental physics of particles \cite{peskin_introduction_1995,weinberg_quantum_1996,weinberg_quantum_1996-1,weinberg_quantum_2000} to the effective description of many body interacting quantum systems such as magnets and dilute atomic gases \cite{fradkin_field_2013}. Thus it is no real exaggeration to say that QFT is the calculus of modern physics \cite{witten_surface_2006,seiberg_nathan_2014}. In contrast to the mature status that calculus enjoys, however, QFT is still far from a stable formulation \cite{howard_georgi_particles_2012,moore_physical_2014,seiberg_nathan_2014} as texts on the subject are not standardised and, further, mathematicians are not yet universally happy with QFT as practiced by physicists. +Quantum field theory (QFT) has become, thanks to Wilson +\cite{wilson_renormalization_1974,wilson_renormalization_1975}, an immensely powerful calculational machine to study a wide variety of physical problems from the fundamental physics of particles \cite{peskin_introduction_1995,weinberg_quantum_1996,weinberg_quantum_1996-1,weinberg_quantum_2000} to many body interacting quantum systems such as magnets and dilute atomic gases \cite{fradkin_field_2013}. It is no exaggeration to say that QFT is the calculus of modern physics \cite{witten_surface_2006,seiberg_nathan_2014}. In contrast to the mature status that calculus enjoys, however, QFT is still far from a stable formulation \cite{howard_georgi_particles_2012,moore_physical_2014,seiberg_nathan_2014} as texts on the subject are not standardised and, further, mathematicians are not yet universally happy with QFT as practiced by physicists. -Amongst the many formulations of QFT, a most popular one is in terms of \emph{lagrangians}. Here one begins with a set of \emph{classical} equations of motion, encapsulated by a lagrangian via the principle of least action \cite{arnold_mathematical_1989}, and then one seeks a \emph{quantisation} of these equations of motion, typically via the path integral prescription. This approach has led to great progress: for example, it works extremely well in the perturbative setting, where quantum field theory is now on a rather firm footing and, more importantly, it also provides an elegant way to approach the \emph{nonperturbative} setting where, e.g., it serves as the basis of lattice gauge theory \cite{creutz_quarks_1985,wilson_confinement_1974}. In the case of lattice gauge theory a dramatic validation of the path integral formulation was recently obtained when the hadronic spectrum of QCD was numerically obtained from first principles \cite{durr_ab_2008}. However, despite the power and ubiquity of the lagrangian/path integral approach, there are still many mysteries concerning nonperturbative QFT. +Amongst the many formulations of QFT, a most popular one is in terms of \emph{lagrangians}. Here one begins with a set of \emph{classical} equations of motion, encapsulated by a lagrangian via the principle of least action \cite{arnold_mathematical_1989}, and then seeks a \emph{quantisation} of these equations of motion, typically via the path integral prescription. This approach has led to great progress: for example, it works extremely well in the perturbative setting, where quantum field theory is on firm footing and, more importantly, it also provides an elegant way to approach the \emph{nonperturbative} setting where, e.g., it serves as the basis of lattice gauge theory \cite{creutz_quarks_1985,wilson_confinement_1974}. In the case of lattice gauge theory a dramatic validation of the path integral formulation was recently obtained when the hadronic spectrum of QCD was numerically obtained from first principles \cite{durr_ab_2008}. However, despite the power and ubiquity of the lagrangian/path integral approach, there are still many mysteries concerning nonperturbative QFT. -One way to make progress on understanding nonperturbative QFT might be to eschew the lagrangian parametrisation altogether. This is not a new idea: in the '60s and '70s the idea of deriving all of physics from the analyticity properties \cite{eden_analytic_2002} of the $S$ matrix was very popular. This idea lost steam\footnote{This is actually not fair: the $S$ matrix bootstrap programme was one of the starting points for string theory, which has subsequently enjoyed tremendous success in understanding QFT, especially via dualities. This is a vast topic which lies beyond the scope of these notes. Here we largely focus on non-string approaches to QFT.} in the late '70s in the wake of the stunning success of the standard model. Recently, however, there has been an upswing in interest in formulations of QFT without the lagrangian. A major impetus here comes from string theory \cite{moore_physical_2014,howard_georgi_particles_2012} where there are arguments that there exists certain quantum field theories \cite{witten_comments_1995,moore_lecture_2012} with no known, or no unique, lagrangian. Additional motivation for studying QFT without lagrangians has been powerfully articulated by Arkani-Hamed and collaborators in the course of the programme to understanding the scattering amplitudes for $\mathcal{N}=4$ supersymmetric Yang-Mills theory \cite{arkani-hamed_what_2010,arkani-hamed_into_2014,arkani-hamed_tree_2008,arkani-hamed_scattering_2012,arkani-hamed_all-loop_2011,arkani-hamed_amplituhedron_2014,arkani-hamed_s-matrix_2010,arkani-hamed_what_2010}: here the core motivation is to find a parametrisation of QFT which exposes hidden symmetries at the expense of manifest unitarity and locality, i.e., to allow spacetime to be an emergent property. +One way to make progress on understanding nonperturbative QFT might be to eschew the lagrangian parametrisation altogether. This is not a new idea: in the '60s and '70s the idea of deriving all of physics from the analyticity properties \cite{eden_analytic_2002} of the $S$ matrix was very popular. This idea lost steam\footnote{This is actually not fair: the $S$ matrix bootstrap programme was one of the starting points for string theory, which has subsequently enjoyed tremendous success in understanding QFT, especially via dualities. This is a vast topic which lies beyond the scope of these notes. Here we largely focus on non-string approaches to QFT.} in the late '70s in the wake of the stunning success of the standard model. Recently, however, there has been an upswing in interest in formulations of QFT without the lagrangian. A major impetus here comes from string theory \cite{moore_physical_2014,howard_georgi_particles_2012} where there are arguments that there exist certain quantum field theories \cite{witten_comments_1995,moore_lecture_2012} with no known, or no unique, lagrangian. Additional motivation for studying QFT without lagrangians has been powerfully articulated by Arkani-Hamed and collaborators in the course of a programme to understand the scattering amplitudes for $\mathcal{N}=4$ supersymmetric Yang-Mills theory \cite{arkani-hamed_what_2010,arkani-hamed_into_2014,arkani-hamed_tree_2008,arkani-hamed_scattering_2012,arkani-hamed_all-loop_2011,arkani-hamed_amplituhedron_2014,arkani-hamed_s-matrix_2010,arkani-hamed_what_2010}: here the core motivation is to find a parametrisation of QFT which exposes hidden symmetries at the expense of manifest unitarity and locality, i.e., to allow spacetime to be an emergent property. -The theme of emergent spacetime plays a crucial role in discussions of the \emph{holographic principle} \cite{bousso_holographic_2002}. In the specific context of the AdS/CFT correspondence \cite{maldacena_large_1999} we have seen that spacetimes of certain associated bulk degrees of freedom are encoded into the hilbert spaces of strongly interacting quantum many body systems living on spatial boundaries. The nature of this encoding is still not completely understood, particularly away from the large-$N$ limit where semiclassical arguments are no longer valid. However, this idea has proved to be so deep that even just taking aim at its general direction has lead to spectacular and exciting progress. Most relevant for this paper is a line of enquiry beginning with the work of Swingle \cite{swingle_entanglement_2012} applying tensor networks to quantify the nature of, and the correspondence between, bulk and boundary degrees of freedom \cite{nozaki_holographic_2012,ryu_aspects_2006,ryu_holographic_2006}. Also closely related, are studies exploiting quantum information theoretic ideas, particularly from the study of quantum error correcting codes \cite{almheiri_bulk_2014} and quantum Shannon theory \cite{czech_information_2014}, aimed at elucidating the interpretation of AdS/CFT duality. These recent studies usually work by first discretisating the problem and then deploying the apparatus of quantum information theory to the resulting discrete system. The precise way in which results obtained in this way survive the limit to the continuum is rather subtle. +The theme of emergent spacetime plays a crucial role in discussions of the \emph{holographic principle} \cite{bousso_holographic_2002}. In the specific context of the AdS/CFT correspondence \cite{maldacena_large_1999} we have seen that spacetimes of certain associated bulk degrees of freedom are encoded in the hilbert spaces of strongly interacting quantum many body systems living on spatial boundaries. The nature of this encoding is still not completely understood, particularly away from the large-$N$ limit where semiclassical arguments are no longer valid. However, this idea has proved to be so deep that even just taking aim in its general direction has lead to spectacular and exciting progress. Most relevant for this paper is a line of enquiry beginning with the work of Swingle \cite{swingle_entanglement_2012} applying tensor networks to quantify the nature of, and the correspondence between, bulk and boundary degrees of freedom \cite{nozaki_holographic_2012,ryu_aspects_2006,ryu_holographic_2006}. Also closely related, are studies exploiting quantum information theoretic ideas, particularly from the study of quantum error correcting codes \cite{almheiri_bulk_2014,pastawski_holographic_2015} and quantum Shannon theory \cite{czech_information_2014}, aimed at elucidating the interpretation of AdS/CFT duality. These recent studies usually work by first discretisating the problem and then deploying the apparatus of quantum information theory to the resulting discrete system. The precise way in which results obtained in this way survive the limit to the continuum is rather subtle. -Continuing with the themes of holography, emergent spacetime geometry, and quantum information theory, there has been a recent flurry of activity centred around the ``firewall'' paradox, initiated by the work of \cite{almheiri_black_2013}. The ensuing debate has prompted many intriguing and original ideas aimed at resolving the paradox. One extremely suggestive proposal \cite{maldacena_cool_2013}, known as ``ER=EPR'', posits that the fabric of spacetime itself is none other than quantum entanglement. This is heady stuff! However it is hard, especially for the quantum information theorist, to make concrete sense of it, especially since at first sight, the proposal appears to be a category error. +Continuing with the themes of holography, emergent spacetime geometry, and quantum information theory, there has been a recent flurry of activity centred around the ``firewall'' paradox, initiated by the work of \cite{almheiri_black_2013}. The ensuing debate has prompted many intriguing and original ideas aimed at resolving the paradox. One extremely suggestive proposal \cite{maldacena_cool_2013}, known as ``ER=EPR'', posits that the fabric of spacetime itself is none other than quantum entanglement. This is heady stuff! However it is hard, especially for the quantum information theorist, to make concrete sense of it, especially since at first sight the proposal appears to be a category error. -We believe that quantum information theorists have interesting things to contribute to high energy physics and see several possibilities. The most direct way would be to develop ideas and results that have already proved successful in the study of strongly correlated complex quantum systems to apply to settings of direct interest in high energy physics. In condensed matter physics inspiration from quantum information theory has led to the development of new variational families of \emph{tensor network states} (TNS), including, the \emph{projected entangled pair states} (PEPS) \cite{verstraete_renormalization_2004} and the \emph{multiscale entanglement renormalisation ansatz} (MERA) \cite{vidal_entanglement_2007,vidal_class_2008}. The crucial idea underlying these developments is that TNS provide a parsimonious and expressive \emph{data structure} to parametrise the hilbert space of physical states naturally arising \cite{poulin_quantum_2011} in local quantum systems \cite{orus_practical_2014, haegeman_geometry_2014,osborne_simulating_2007,bravyi_topological_2010,bravyi_short_2011,hastings_lieb-schultz-mattis_2004,hastings_area_2007,osborne_efficient_2006}. One way to carry out the goal of understanding quantum fields via tensor network methods is to formulate TNS directly in the continuum. This approach has already given rise to the continuous matrix products states (cMPS) \cite{verstraete_continuous_2010,osborne_holographic_2010, haegeman_calculus_2013}, continuous PEPS \cite{jennings_variational_2012}, and continuous MERA classes \cite{haegeman_entanglement_2013}. Such continuous TNS have provided some new insights in the study of some problems in high-energy physics, and promise to provide a powerful way to reason about entangled quantum fields. Another way, adopted in this paper, is to simply understand how discrete TNS approximate a given QFT. This approach is easier to implement numerically, and also more directly allows the computation of, e.g., quantum entanglement. +We believe that quantum information theorists have interesting things to contribute to high energy physics, and see several possible avenues forward. The most direct way would be to develop ideas and results that have already proved successful in the study of strongly correlated complex quantum systems to apply to settings of direct interest in high energy physics. In this context, new variational families of \emph{tensor network states} (TNS), including, the \emph{projected entangled pair states} (PEPS) \cite{verstraete_renormalization_2004} and the \emph{multiscale entanglement renormalisation ansatz} (MERA) \cite{vidal_entanglement_2007,vidal_class_2008} could be exploited. The crucial idea underlying these developments is that TNS provide a parsimonious and expressive \emph{data structure} to parametrise the hilbert space of physical states naturally arising \cite{poulin_quantum_2011} in local quantum systems \cite{orus_practical_2014, haegeman_geometry_2014,osborne_simulating_2007,bravyi_topological_2010,bravyi_short_2011,hastings_lieb-schultz-mattis_2004,hastings_area_2007,osborne_efficient_2006}. One way to carry out the goal of understanding quantum fields via tensor network methods is to formulate TNS directly in the continuum. This approach has already given rise to the continuous matrix products states (cMPS) \cite{verstraete_continuous_2010,osborne_holographic_2010, haegeman_calculus_2013}, continuous PEPS \cite{jennings_variational_2012}, and continuous MERA classes \cite{haegeman_entanglement_2013}. Such continuous TNS have provided some new insights in the study of some problems in high-energy physics, and promise to provide a powerful way to reason about entangled quantum fields. Another way, adopted in this paper, is to simply understand how discrete TNS approximate a given QFT. This approach is easier to implement numerically, and also more directly allows the computation of, e.g., quantum entanglement. -Another avenue where quantum information theory seems likely to lead to progress in high energy physics is by exploiting quantum computers to directly simulate scattering processes. This is an important goal, even in the perturbative setting, because quantum computers allow the computation of scattering amplitudes involving many particles which require the summation of a prohibitive number of Feynman diagrams. Pursuit of this idea has led to the development of discrete quantum simulation algorithms for scalar field theory \cite{jordan_quantum_2012,jordan_quantum_2011} and the Gross-Neveu model on the lattice \cite{jordan_quantum_2014}. Here there are again fascinating questions about how to understand the nonperturbative regime and the approach to the continuum. +Another avenue where quantum information theory seems likely to lead to progress in high energy physics is by exploiting quantum computers to directly simulate scattering processes. This is an important goal, even in the perturbative setting, because quantum computers allow the computation of scattering amplitudes involving many particles requiring the summation of a prohibitive number of Feynman diagrams. Pursuit of this idea has led to the development of discrete quantum simulation algorithms for scalar field theory \cite{jordan_quantum_2012,jordan_quantum_2011} and the Gross-Neveu model on the lattice \cite{jordan_quantum_2014}. Here there are again fascinating questions about how to understand the nonperturbative regime and the approach to the continuum. Finally, and somewhat more speculatively, the work \cite{arkani-hamed_what_2010,arkani-hamed_into_2014,arkani-hamed_tree_2008,arkani-hamed_scattering_2012,arkani-hamed_all-loop_2011,arkani-hamed_amplituhedron_2014,arkani-hamed_s-matrix_2010,arkani-hamed_what_2010} of Arkani-Hamed and collaborators is rather suggestive to someone with a quantum information background: the idea that scattering amplitudes can be directly obtained from the volumes of a certain special convex set known as the \emph{amplituhedron} resonates strongly with themes that have been discussed in the quantum information literature. In particular, the amplituhedron bears a superficial resemblence to the convex set of reduced density operators for (translation invariant) complex quantum systems \cite{verstraete_matrix_2006,zauner_symmetry_2014}. For instance, one can argue that scattering amplitudes for any complex quantum system may be directly derived from knowledge of a related convex set \cite{osborne_tobiasosborne}. Whether the connection between these two topics is more than metaphoric is far from clear, however, it seems like a deep idea worth exploring further. -So it seems that there are a variety of ways in which quantum information theory could contribute to high energy theory. However, there is a fundamental difficulty facing the quantum information theorist, namely that quantum fields involve a continuous infinity of degrees of freedom. This setting is rather far removed from the home ground of quantum information theory, namely, the qubit. When faced with the insecurity of dealing with an unfamiliar setting it is very tempting to find a concrete rigourous mathematical formulation for the theory. In the case of quantum field theory the pursuit of a satisfying mathematical objective has sparked a remarkable variety of different approaches, invoking mathematics from the algebraic, probabilistic, to the geometric. It is not possible to do justice to this work here, but we are lucky in that we can simply refer to an excellent recent survey of Douglas \cite{douglas_foundations_2012}, who provides a broad overview of the existing approaches to a mathematical theory of QFT. It is certainly possible that one way quantum information theory could contribute to high energy physics is via one of the aforementioned mathematical approaches. However, we now end up with two problems, namely, understanding and developing quantum information ideas for a given mathematical approach, and then working out how to apply them to physical problems. +So it seems that there are a variety of ways in which quantum information theory could contribute to high energy theory. However, there is a fundamental difficulty facing the quantum information theorist: quantum fields involve a continuous infinity of degrees of freedom, a setting rather far removed from the home ground of quantum information theory, namely, the qubit. Faced with the insecurity of exploring such unfamiliar territory it is tempting to work from a comfortable rigourous mathematical formulation. However, a satisfying universally accepted mathematical framework for QFT has yet to be found; there are a panoply of existing approaches from the algebraic, probabilistic, to the geometric (we are lucky in that we can simply refer to an excellent recent survey of Douglas \cite{douglas_foundations_2012}, who provides a broad overview of the existing approaches to a mathematical theory of QFT). It is certainly possible that quantum information theory could contribute to high energy physics via one of the aforementioned mathematical approaches. However, we now end up with two problems, namely, understanding and developing quantum information ideas for a given mathematical approach, and then working out how to apply them to physical problems. -The approach adopted in these notes is rather different. Here we advocate directly understanding QFT as practiced by physicists from a quantum information perspective. This differs in two important ways from existing mathematical approaches that have been so far developed. Firstly, we regard QFT as an effective theory and, secondly, we focus on quantum state space in addition to quantum observables. By simply understanding the way QFT is formulated as an effective theory, we can avoid a lot of heavy mathematical machinery and get down to concrete problems and calculations. +The approach adopted in these notes is rather different. Here we advocate directly understanding QFT as practiced by physicists. This differs in two important ways from most of the existing mathematical approaches that have been so far developed. Firstly, we regard QFT as an \emph{effective theory} and, secondly, we focus on quantum \emph{state space} in addition to quantum observables. By understanding the way QFT is formulated as an effective theory, we can avoid a lot of heavy mathematical machinery and get down to concrete problems and calculations. -So what is a quantum field state? Our answer, as we argue in the course of this paper, is that a quantum field state is simply a \emph{sequence of states} of discretised theories with a certain property, namely, that each term gets closer to each other (i.e., a \emph{Cauchy sequence}). This is directly analogous to how we study, via computer, classical fields, like fluids and gases and the electromagnetic field. Following in the footsteps of Wilson, we spend some time discussing how to measure ``closeness'', leading to the introduction of a family of quantum information distance measures quantifying the large-scale behaviour of a many particle state. +So what is a quantum field state? Our answer is that it is a \emph{sequence of states} of discretised theories with a certain property, namely, that each term gets closer to each other (i.e., a \emph{Cauchy sequence}). This is directly analogous to how we study, via computer, classical fields, like fluids and gases and the electromagnetic field. Following in the footsteps of Wilson, we spend some time discussing how to measure ``closeness'', leading to the introduction of a family of quantum information distance measures quantifying the large-scale behaviour of a quantum many body state. -We've structured this paper as follows. We begin in \S\ref{sec:opqp} with a short overview of operational quantum mechanics with an emphasis on density operators, completely positive maps, and POVMs. After setting up this basic language we move on in \S\ref{sec:quant} to a short overview of the problems of quantisation. Then we discuss in \S\ref{sec:whatisqft}, on a purely heuristic level, what a field theory really ought to be. With this motivation we then discuss effective theories in \S\ref{sec:effectivetheories}, which directly leads to the Wilsonian formulation of effective quantum field theory, reviewed in \S\ref{sec:wilson}. This discussion is then used as the direct motivation for our definition of effective quantum field states in \S\ref{sec:effectiveqftstates}. The construction of quantum field states via completion is then introduced in \S\ref{sec:qftcompletion}. The explanation of the renormalisation group as a means to construct Cauchy sequences of states is then described in \S\ref{sec:cauchyseqrg}. Finally, we conclude with some discussion and outlook in \S\ref{sec:discussion}. +We've structured this paper as follows. We begin in \S\ref{sec:opqp} with a short overview of operational quantum mechanics with an emphasis on density operators, completely positive maps, and POVMs. After setting up this basic language we move on in \S\ref{sec:quant} to a short overview of the problems of quantisation. Then we discuss in \S\ref{sec:whatisqft}, on a purely heuristic level, what a field theory really ought to be. With this motivation in hand we then discuss effective theories in \S\ref{sec:effectivetheories}, which directly leads to the Wilsonian formulation of effective quantum field theory, reviewed in \S\ref{sec:wilson}. This discussion is then used as the direct motivation for our definition of effective quantum field states in \S\ref{sec:effectiveqftstates}. The construction of quantum field states via completion is then introduced in \S\ref{sec:qftcompletion}. The explanation of the renormalisation group as a means to construct Cauchy sequences of states is then described in \S\ref{sec:cauchyseqrg}. Finally, we conclude with some discussion and outlook in \S\ref{sec:discussion}. \section{Operational quantum physics}\label{sec:opqp} -Throughout these notes we emphasise the \emph{operational} or \emph{modular}\footnote{Modular refers, in this context, to the idea that the primitive operations of preparation, evolution, and measurement may be composed arbitrarily to build \emph{quantum circuits}, much as we do in building classical circuits.} \emph{viewpoint}: here the focus is on physical quantities with an \emph{operational interpretation}, i.e., a physical quantity is considered operationally meaningful only if there exists, at least in principle, an experiment which could measure it. The modular viewpoint is common within quantum information theory as it lends itself very naturally to quantum circuits. The operational view also seems to mesh rather well with the Wilsonian view of QFT, where QFT is seen as an effective theory. +Throughout these notes we emphasise the \emph{operational} or \emph{modular}\footnote{Modular refers, in this context, to the idea that the primitive operations of preparation, evolution, and measurement may be composed arbitrarily to build \emph{quantum circuits}, much as we do in building classical circuits.} \emph{viewpoint}: here the focus is on physical quantities with an \emph{operational interpretation}, i.e., a physical quantity is considered operationally meaningful only if there exists, at least in principle, an experiment which could measure it. The modular viewpoint is common within the quantum information community as it lends itself very naturally to the discussion of quantum circuits. The operational view also seems to mesh rather well with the Wilsonian view of QFT, where QFT is seen as an effective theory. -A convenient way to discuss quantum physics within the operational or modular viewpoint is via \emph{observables and effects}, \emph{density operators}, and \emph{completely positive maps} \cite{ludwig_foundations_1983,davies_quantum_1976}. This language may be unfamiliar to the reader and we pause a moment to review it here. Firstly, the way we characterise \emph{quantum} (indeed, also \emph{classical}) systems is via a set $\mathcal{A}$ of \emph{observables}. The observables typically form an algebra isomorphic to the bounded operators $\mathcal{B}(\mathcal{H})$ on some hilbert space $\mathcal{H}$, however this is not strictly necessary\footnote{The algebra structure is usually employed as a proxy for the positivity notion. All that we actually need is that $\mathcal{A}$ is an \emph{Archmidean Order Unit} (AOU) vector space \cite{paulsen_vector_2009,kleinmann_typical_2013}.} for there to be a probability interpretation. For the probability interpretation we need only require that $\mathcal{A}$ has a notion of \emph{positivity}, i.e., there is a cone $\mathcal{A}^+ \subset \mathcal{A}$ of positive elements and that there is a distinguished unit element $\mathbb{I}\in\mathcal{A}$ which is also positive. A good example to keep in mind here is that of the \emph{qubit}, where $\mathcal{A}\equiv M_2(\mathbb{C})$, the algebra of $2\times 2$ complex matrices and $\mathcal{A}^+ \equiv \{M\in \mathcal{A}\,|\, M\ge 0\}$ (with $\ge$ denoting the positive semidefinite order, i.e., $M\ge 0$ if and only if there exists $A\in \mathcal{A}$ such that $M = A^\dag A$). Another example is that of a \emph{classical} system, which is characterised by a \emph{commutative algebra} $\mathcal{A}\equiv \mathcal{C}(X)$, the set of functions from some set $X$ to $\mathbb{C}$; a \emph{classical bit} corresponds to the choice $X \equiv \{0,1\}$. An \emph{effect} $E\in\mathcal{A}^+$ is then what we call an observable corresponding to an \emph{outcome}, \emph{proposition}, \emph{predicate}, or \emph{yes/no measurement}. It is characterised by the property that $0\le E\le \mathbb{I}$. The unit element is the effect corresponding to empty predicate, that is, no assertion. For the qubit example above the projector $P_0 = \left(\begin{smallmatrix} 1 & 0 \\ 0 & 0\end{smallmatrix}\right)$ is the effect corresponding to the assertion that the system is in the ``zero'' configuration. A POVM\footnote{The acronym POVM stands for ``positive operator valued measure'', which takes its full meaning when the observable can take a continuous sets of values. } corresponds to a measurement of a system and---provided it can take only finitely many values---comprises of a set $\mathcal{M} = \{E_j\}_{j=1}^n$ of effects such that +A convenient way to discuss quantum physics within the operational or modular viewpoint is via \emph{observables and effects}, \emph{density operators}, and \emph{completely positive maps} \cite{ludwig_foundations_1983,davies_quantum_1976}. This language may be unfamiliar to the reader and we pause a moment to review it here. Firstly, the way we characterise \emph{quantum} (indeed, also \emph{classical}) systems is via a set $\mathcal{A}$ of \emph{observables}. The observables typically form an algebra isomorphic to the bounded operators $\mathcal{B}(\mathcal{H})$ on some hilbert space $\mathcal{H}$, although this is not necessary\footnote{The algebra structure is usually employed as a proxy for \emph{positivity}. All that we actually need is that $\mathcal{A}$ is an \emph{Archmidean Order Unit} (AOU) vector space \cite{paulsen_vector_2009,kleinmann_typical_2013}.} for there to be a probability interpretation. For the probability interpretation we need only require that $\mathcal{A}$ has a notion of \emph{positivity}, i.e., there is a cone $\mathcal{A}^+ \subset \mathcal{A}$ of positive elements and that there is a distinguished unit element $\mathbb{I}\in\mathcal{A}$ which is also positive. A good example to keep in mind here is that of the \emph{qubit}, where $\mathcal{A}\equiv M_2(\mathbb{C})$, the algebra of $2\times 2$ complex matrices and $\mathcal{A}^+ \equiv \{M\in \mathcal{A}\,|\, M\ge 0\}$ (with $\ge$ denoting the positive semidefinite order, i.e., $M\ge 0$ if and only if there exists $A\in \mathcal{A}$ such that $M = A^\dag A$). Another example is that of a \emph{classical} system, which is characterised by a \emph{commutative algebra} $\mathcal{A}\equiv \mathcal{C}(X)$, the set of functions from some set $X$ to $\mathbb{C}$; a \emph{classical bit} corresponds to the choice $X \equiv \{0,1\}$. An \emph{effect} $E\in\mathcal{A}^+$ is then what we call an observable corresponding to an \emph{outcome}, \emph{proposition}, \emph{predicate}, or \emph{yes/no measurement}. It is characterised by the property that $0\le E\le \mathbb{I}$. The unit element is the effect corresponding to the empty predicate, that is, no assertion. For the qubit example above the projector $P_0 = \left(\begin{smallmatrix} 1 & 0 \\ 0 & 0\end{smallmatrix}\right)$ is the effect corresponding to the assertion that the system is in the ``zero'' configuration. A POVM\footnote{The acronym POVM stands for ``positive operator valued measure'', which takes its full meaning when the observable can take a continuous sets of values. } corresponds to a measurement of a system and---provided it can take only finitely many values---comprises of a set $\mathcal{M} = \{E_j\}_{j=1}^n$ of effects such that \begin{equation} \sum_{j=1}^n E_j = \mathbb{I}. \end{equation} -The subscript label $j$ is what carries the information about what \emph{outcome} was observed when the measurement took place. It can be interpreted directly as the \emph{value} of the observable being measured, or simply as a label of that value. It is worth stressing that the effects $E_j$ need not be projections. +The subscript label $j$ is what carries the information about what \emph{outcome} was observed when the measurement took place. It can be interpreted directly as the \emph{value} of the observable being measured, or simply as a label of that value. It is worth stressing that the effects $E_j$ need not be projections. Traditionally we speak of hermitian operators as observables in quantum mechanics, but what is actually meant when we declare that the hermitian operator $M = \sum_{j=1}^n m_j E_j$ is an ``observable'' is that we implement the corresponding POVM $\mathcal{M} = \{E_j\}_{j=1}^n$, where $E_j$ are the spectral projections for $M$. The expectation value $\langle M \rangle$ corresponds to the first moment of the probability distribution determined by $\mathcal{M}$. -Compositions of systems is described via the \emph{tensor product} operation: suppose we have two systems $A$ and $B$ characterised by the observable sets $\mathcal{A}_A$ and $\mathcal{A}_B$, respectively. The joint system $AB$ is then characterised by the observable set $\mathcal{A}_{AB} \equiv \mathcal{A}_A\otimes \mathcal{A}_B$. This space allows the simultaneous observation of effects in both $\mathcal{A}_A$ \emph{and} $\mathcal{A}_B$. The \emph{classical composition} of two systems $\mathcal{A}_A$ and $\mathcal{A}_B$, where we allow the observations of effects in \emph{either} $\mathcal{A}_A$ \emph{or} $\mathcal{A}_B$, i.e., our system is either of one type or another -- superpositions are not allowed -- is described by the \emph{direct sum} operation $\mathcal{A}_{A}\oplus \mathcal{A}_B$. This is the smallest space of effects allowing us to probabilistically measure an effect from $\mathcal{A}_A$ or $\mathcal{A}_B$ +Quantum composition of systems is described via the \emph{tensor product} operation: suppose we have two systems $A$ and $B$ characterised by the observable sets $\mathcal{A}_A$ and $\mathcal{A}_B$, respectively. The joint system $AB$ is then characterised by the observable set $\mathcal{A}_{AB} \equiv \mathcal{A}_A\otimes \mathcal{A}_B$. This space allows the simultaneous observation of effects in both $\mathcal{A}_A$ \emph{and} $\mathcal{A}_B$. The \emph{classical composition} of two systems $\mathcal{A}_A$ and $\mathcal{A}_B$, where we allow the observations of effects in \emph{either} $\mathcal{A}_A$ \emph{or} $\mathcal{A}_B$, i.e., our system is either of one type or another, is described by the \emph{direct sum} operation $\mathcal{A}_{A}\oplus \mathcal{A}_B$. This is the smallest space of effects allowing us to probabilistically measure an effect from $\mathcal{A}_A$ or $\mathcal{A}_B$. A \emph{state} $\omega:\mathcal{A}\rightarrow \mathbb{C}$ on the observable set $\mathcal{A}$ is a \emph{positive}, \emph{normalised}, and \emph{linear} functional, i.e., $\omega:\mathcal{A}^+\rightarrow \mathbb{R}^+$ and $\omega(e) = 1$. A state describes a \emph{preparation} of the system, and captures all the information relevant for the statistical outcomes of measurements on the system. The probability $p_E$ that, after a measurement of a POVM $\mathcal{M}$, an outcome with corresponding effect $E$ occurs is given by $p_E = \omega(E)$. Usually in quantum information theory we work with finite-dimensional quantum systems with $\mathcal{A}\equiv M_d(\mathbb{C})$ so that we can represent states via \emph{density operators} $\rho\in M_d(\mathbb{C})$ according to $\omega(M) \equiv \tr(\rho M)$, with $\tr(\rho) = 1$, $\rho \ge 0$. (Be aware that in infinite-dimensional settings it is not always possible to find a density operator corresponding to a state as the trace condition can easily fail.) A state $\omega$ is \emph{pure} if it cannot be written as a convex combination of other states, i.e., if $\omega \not= p \omega' + (1-p) \omega''$, with $p\in (0,1)$. Continuing the qubit example from above we see that single-qubit states $\omega$ correspond to $2\times 2$ density operators \begin{equation} @@ -112,7 +112,7 @@ \section{Operational quantum physics}\label{sec:opqp} \end{theorem} Here $V$ describes both the unitary dynamics and the reduction to a subsystem and $X\mapsto X\otimes \mathbb{I}$ describes the adjunction of the ancillary system. -There are several key examples of CP maps naturally arising in physics. The first example are the so-called \emph{cq channels} (cq = classical-quantum), describing \emph{preparations}: these are channels from a set of quantum observables $\mathcal{A}$ to classical observables $\mathcal{C}(X)$ with $X \equiv \{1,2,\ldots, n\}$, +There are several key examples of CP maps naturally arising in physics. The first examples are the so-called \emph{cq channels} (cq = classical-quantum), describing \emph{preparations}: these are channels from a set of quantum observables $\mathcal{A}$ to classical observables $\mathcal{C}(X)$ with $X \equiv \{1,2,\ldots, n\}$, \begin{equation} \mathcal{E}(E) \equiv \sum_{j = 1}^n \omega_j (E) \delta_j, \end{equation} @@ -132,7 +132,7 @@ \section{Operational quantum physics}\label{sec:opqp} \begin{center} \includegraphics{prepevolvemeasure.pdf} \end{center} -Here, conditioned on a classical input with the value $j$ a quantum state $\rho_j$ is prepared. This is subsequently evolved according to a completely positive map $\mathcal{E}$. Finally a POVM measurement $\mathcal{M} = \{E_k\}$ is performed producing the classical output $k$. Although this modular view of quantum mechanics won't be directly exploited in the sequel, it is present in our minds when we come to discussing the building blocks of quantum field theory. +Here, conditioned on a classical input with the value $j$, a quantum state $\rho_j$ is prepared. This is subsequently evolved according to a completely positive map $\mathcal{E}$. Finally a POVM measurement $\mathcal{M} = \{E_k\}$ is performed producing the classical output $k$. Although this modular view of quantum mechanics won't be directly exploited in the sequel, it is present in our minds when we come to discussing the building blocks of quantum field theory. Thus, from now on, when we say the word ``theory'' we take this to mean the specification of a triple $(\mathcal{A}, \mathcal{E}_t, \omega)$ of an observable set $\mathcal{A}$, a family of CP maps $\mathcal{E}_t: \mathbb{R}\times\mathcal{A}\rightarrow \mathcal{A}$ of possible evolutions, and a preparation $\omega$. We think of $\mathcal{A}$ as the space of \emph{equal-time} observables. The channel $\mathcal{E}_t$ is what implements the operation of translation in \emph{time}\footnote{In the case where the theory admits an action of a larger group $G$ of, say, spacetime translations, or Poincar\'e transformations, we then suppose that $\mathcal{E}_g$ is indexed by elements of $g\in G$.}. It is convenient to exploit the shorthand notation $A(t) \equiv \mathcal{E}_t(A)$. Thus we can now discuss observables corresponding to measurements at different times. In the case where $\mathcal{A}$ has an algebraic structure\footnote{In the case where $\mathcal{A}$ does not have an algebraic structure we have to explicitly describe correlation functions via instruments.} this allows us to introduce the observables corresponding to $n$-point correlation functions, namely, @@ -147,15 +147,15 @@ \section{Operational quantum physics}\label{sec:opqp} Note that a ``standard'' $n$-point correlation function $\langle A_n(t_n) \cdots A_2(t_2)A_1(t_1) \rangle$ is not directly measurable in quantum mechanics as, in general, the product $A_n(t_n) \cdots A_2(t_2)A_1(t_1)$ is not even hermitian. Instead, such correlators must be inferred from scattering \cite{taylor_scattering_2006} or interference experiments \cite{glauber_quantum_1963,mandel_optical_1995}. \section{Quantisation isn't a mystery, it's an inverse problem}\label{sec:quant} -Before we get started with understanding quantum field states we pause a moment to stress a simple yet important point. The universe didn't become quantum in 1927 at the Fifth Solvay International Conference, it has always been quantum. The reason that we didn't notice quantum effects for such a long time is because of \emph{decoherence} \cite{joos_decoherence_2003,gardiner_quantum_2010}, i.e., the unavoidable loss of quantum coherence due to uncontrolled interactions with unobservable environment degrees of freedom. In the presence of quantum noise pure unitary dynamics described by a unitary channel $\mathcal{U}_t$ obeying +Before we get started with quantum field states we pause a moment to stress a simple yet important point: the universe didn't become quantum in 1927 at the Fifth Solvay International Conference, it has always been quantum. The reason that we didn't notice quantum effects for such a long time is because of \emph{decoherence} \cite{joos_decoherence_2003,gardiner_quantum_2010}, i.e., the unavoidable loss of quantum coherence due to uncontrolled interactions with unobservable environment degrees of freedom. In the presence of quantum noise pure unitary dynamics described by a unitary channel $\mathcal{U}_t$ obeying \begin{equation} \frac{d}{dt}\mathcal{U}_t(X) \equiv -i [H, \mathcal{U}_t(X)] \end{equation} -is modified to a noisy CP map $\mathcal{E}_t$ generated by +is modified \cite{davies_quantum_1976} to a noisy CP map $\mathcal{E}_t$ generated by \begin{equation} - \frac{d}{dt}\mathcal{E}_t(X) \equiv -i [H, \mathcal{E}_t(X)] -\frac12\sum_{\alpha=1}^m L_\alpha^\dag L_\alpha \mathcal{E}_t(X) + \mathcal{E}_t(X) L_\alpha^\dag L_\alpha - L_\alpha \mathcal{E}_t(X) L_\alpha^\dag + \frac{d}{dt}\mathcal{E}_t(X) \equiv -i [H, \mathcal{E}_t(X)] -\frac12\sum_{\alpha=1}^m L_\alpha^\dag L_\alpha \mathcal{E}_t(X) + \mathcal{E}_t(X) L_\alpha^\dag L_\alpha - 2 L_\alpha \mathcal{E}_t(X) L_\alpha^\dag \end{equation} -which can, for quantum systems with a continuous degree of freedom (e.g., a particle on the line), be \emph{very effectively} modelled by a symplectic transformation on phase space. +which can, for quantum systems with a continuous degree of freedom (e.g., a particle on the line), usually be \emph{very effectively} modelled by a symplectic transformation on a classical phase space. \begin{center} \includegraphics{Decoherence.pdf} \end{center} @@ -165,18 +165,16 @@ \section{Quantisation isn't a mystery, it's an inverse problem}\label{sec:quant} \begin{center} \includegraphics{Quantisation.pdf} \end{center} -From this perspective it isn't so surprising that quantisation prescriptions aren't universal maps between classical systems and quantum systems, i.e., in mathematical language, \emph{functors}. This is an important observation because it is the first serious sign that there is some room to play with in finding quantum field theories: if we are looking for a quantum system with a specific effective classical description in the presence of decoherence there are many answers that will lead to equivalent results, giving us more room to find one with a useful parsimonious description. +From this perspective it isn't so surprising that quantisation prescriptions aren't universal maps between classical systems and quantum systems, i.e., in mathematical language, \emph{functors}. -So how is the inverse problem of quantisation solved? A vitally important role guiding us toward a solution is played by \emph{symmetries}: if a desired classical limit is invariant under some group of symmetry operations then it is reasonable to assume that a good quantisation ought to furnish some representation of the same symmetry group (especially if the envisaged decoherence process leading to the classical limit is not expected to break the symmetry). This radically cuts down the search space we need to cover in looking for a quantisation. It can turn out, however, that the full symmetry group cannot be represented on a given quantisation, in which case we say that one or more symmetries are \emph{anomolous}. It is an interesting question whether, in general, anomolies disappear in the classical limit under a reasonable model of decoherence. +So how is the inverse problem of quantisation solved? A vitally important role guiding us toward a solution is played by \emph{symmetries}: if a desired classical limit is invariant under some group of symmetry operations then it is reasonable to assume that a good quantisation ought to furnish some representation of the same symmetry group (especially if the envisaged decoherence process leading to the classical limit is not expected to break the symmetry). This radically cuts down the search space we need to cover in looking for a quantisation. It can turn out, however, that the full symmetry group cannot be represented in a given quantisation, in which case we say that one or more symmetries are \emph{anomolous}. It is an interesting question whether, in general, anomolies disappear in the classical limit under a reasonable model of decoherence. In the more modular language promoted in these notes we simply simplistically regard decoherence as a channel $\mathcal{D}$ that gets applied to our system before we perform our measurements. Thus, in the heisenberg picture, you can think of decoherence as modifying the effects we can measure to more noisy effects. \section{What is a field theory}\label{sec:whatisqft} -Let's now begin our discussion proper by contemplating, at a purely heuristic level, what a \emph{field theory} should be. On a purely intuitive level, a \emph{field} (either quantum or classical) is supposed to comprise of \emph{continuously} many degrees of freedom, i.e., roughly speaking, there is a degree of freedom for each point in space $\mathbb{R}^d$ (or spacetime $\mathbb{R}\times \mathbb{R}^d$). When dealing with such a vast abundance of degrees of freedom the task of just specifying a state of such a field becomes deeply nontrivial. +We begin our discussion by contemplating, at a purely heuristic level, what a \emph{field theory} should be. On a purely intuitive level, a \emph{field} (either quantum or classical) comprises \emph{continuously} many degrees of freedom, i.e., roughly speaking, there is a degree of freedom for each point in space $\mathbb{R}^d$ (or spacetime $\mathbb{R}\times \mathbb{R}^d$). When dealing with such a vast abundance of degrees of freedom the task of just specifying a state of such a field becomes deeply nontrivial. -Classically, this task can be largely thought of as being solved by calculus. Here \emph{pure field states} can be simply defined to be continuous functions $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$. The space $C(\mathbb{R}^D)$ (or, in the case of spacetime, $C(\mathbb{R}\times\mathbb{R}^D)$) of all \emph{mathematically} possible such field states is rather wild: it is uncountably infinite. Worse, the task of understanding probability measures on such a space is deeply nontrivial, meaning that a statistical theory of classical fields is already extremely hard. - -Of course, not all of the states in $C(\mathbb{R}^D)$ are meant to be physically realisable, i.e., we haven't yet specified the conditions a \emph{physical state} must satisfy. Classically this is done by requiring that physical pure states satisfy certain differential equations. For example, in a $(1+1)$-dimensional spacetime of points $(t,x)\in \mathbb{R}^2$ we could demand that valid physical states satisfy +Classically, this task is largely solved by calculus. Here \emph{pure field states} can be simply defined to be continuous functions $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$. The space $C(\mathbb{R}^D)$ (or, in the case of spacetime, $C(\mathbb{R}\times\mathbb{R}^D)$) of all \emph{mathematically} possible such field states is rather wild -- it contains fractal monsters -- but not all of the states in $C(\mathbb{R}^D)$ are meant to be #emph{physically realisable}. Classically we specify physical pure states by requiring that they satisfy certain differential equations. For example, in a $(1+1)$-dimensional spacetime of points $(t,x)\in \mathbb{R}^2$ we could demand that valid physical states satisfy \begin{equation} \frac{\partial^2\phi}{\partial t^2} - \frac{\partial^2\phi}{\partial x^2} + m^2\phi = 0. \end{equation} @@ -184,7 +182,7 @@ \section{What is a field theory}\label{sec:whatisqft} \begin{center} \includegraphics{difffunc.pdf} \end{center} -While this doesn't help us solve the problem of building statistical theories of classical fields, it does at least allow us to tame the problem of understanding pure states and their dynamics for systems of continuously many classical degrees of freedom. +While this doesn't help us solve the problem of building statistical theories of classical fields -- the task of understanding probability measures on infinite dimensional spaces is deeply nontrivial -- it does at least allow us to tame the problem of understanding pure states and their dynamics for systems of continuously many classical degrees of freedom. But what about quantum theories? Here we encounter a fundamentally new problem not present in the classical case: it is now hard to even define \emph{pure} field states. Naively this should be straightforward: just define the space of pure states to be the tensor product \begin{equation} @@ -194,13 +192,13 @@ \section{What is a field theory}\label{sec:whatisqft} \begin{equation} \{|\phi\rangle \}_{\phi:\mathbb{R}^d\rightarrow \mathbb{R}}. \end{equation} -While this initially looks reasonable we quickly see that there is a new difficulty: what superpositions are we going to allow? All of them? Surely not: we must find equations that specify for us the physically realisable states. Here we can no longer take recourse to calculus for help. Indeed, the problem of specifying physically realisable pure states of quantum fields is intimately tied to the problem of writing probability measures for classical fields in one lower spatial dimension via the so called ``classical-quantum'' correspondence where the path integral for a system in $D$ spatial dimensions can be regarded, via Wick rotation, as defining a statistical mechanical system in $D+1$ euclidean dimensions. +While this initially looks reasonable we quickly find a new problem: what superpositions are we going to allow? All of them? Surely not: we must find equations that specify for us the physically realisable states. Here we can no longer take recourse to calculus for help. Indeed, the problem of specifying physically realisable pure states of quantum fields is intimately tied to the problem of writing probability measures for classical fields in one lower spatial dimension via the so called ``classical-quantum'' correspondence where the path integral for a system in $D$ spatial dimensions can be regarded, via Wick rotation, as defining a statistical mechanical system in $D+1$ euclidean dimensions. In both the classical statistical field and the quantum field cases we have come up against a fundamental physical problem (as opposed to a technical mathematical problem), namely that of specifying interesting states of fields (in the former case, as probability measures, and in the latter as pure states). What we ideally want is a physical principle that tells what are the ``good'', or \emph{physical}, field states versus the ``bad'', or \emph{unphysical}, states. \section{Effective field theories}\label{sec:effectivetheories} -Suppose we have some extraordinarily complicated system of many particles -- a good example to keep in mind is \emph{water}. Now if it were easy, \emph{at no cost}, for us to make any conceivable measurement on the system allowed by quantum mechanics, then there is \emph{no way} we'd be fooled into thinking water is anything other than a collection of a vast number of fundamental particles, quarks, gluons, etc., in some incredibly complicated evolving entangled state. The reason we don't see water like this is that we \emph{can't} make any measurement of the system without paying some kind of bill: the more complicated the measurement, the more we have to pay. Thus we have to settle with making measurements of simpler quantities. For example, our eyes are basically a pair of pretty crappy photon detectors and thus when we look at a water sample we are simply carrying out a very noisy and inefficient POVM. Now here is the main point: when you only have access to a handful of observables then you can formulate a \emph{simpler hypothesis} which can still explain all the observational data you can obtain. This simpler hypothesis is an \emph{effective theory} for the system. Simpler here can mean many things, but in the context of this paper it is via a field theory\footnote{Why are fields simple? The answer is calculus: it is often easier to calculate integrals than sums.}. +Suppose we have some extraordinarily complicated system of many particles -- a good example to keep in mind is \emph{water}. Now if it were easy, \emph{at no cost}, for us to make any conceivable measurement on the system allowed by quantum mechanics, then there is \emph{no way} we'd be fooled into thinking water is anything other than a collection of a vast number of fundamental particles, quarks, gluons, etc., in some incredibly complicated evolving entangled state. The reason we don't see water like this is that we \emph{can't} make any measurement of the system without paying some kind of bill: the more complicated the measurement, the more we have to pay. Thus we have to settle with making measurements of simpler quantities. For example, our eyes are basically a pair of pretty crappy photon detectors and thus when we look at a water sample we are simply carrying out a very noisy and inefficient POVM. Now here is the main point: when you only have access to fewer observables then you can formulate a \emph{simpler hypothesis} which can still explain all the observational data you can obtain. This simpler hypothesis is an \emph{effective theory} for the system. Simpler here can mean many things, but in the context of this paper it is via a field theory\footnote{Why are fields simple? The answer is calculus: it is often easier to calculate integrals than sums.}. How can we model the large-scale degrees of freedom that we humans with our limited resources can access? One very simplified way is by developing a \emph{zooming out} operation. Since zooming out corresponds to \emph{ignoring information}, this operation should be representable in quantum mechanics as an irreversible CP map $\mathcal{E}$. The reason that it has to be irreversible is that it must prevent us from measuring degrees of freedom that we would otherwise be able to measure: after all, if we could measure all the observables after zooming out that we could measure before then in what sense can we be have said to have zoomed out? In the context of lattice systems there is a very convenient way to implement the zoom-out operation, namely, via Kadanoff blocking. This is the CP map whereby a block of spins is mapped to single spin via the partial trace channel, and then the lattice is rescaled. @@ -225,7 +223,9 @@ \section{The Wilsonian formulation of effective quantum field theory}\label{sec: Hence we now imagine that $\mathcal{A}_{\text{reg}}$ lives inside some even larger space $\mathcal{A}$ of ``all'' observables/effects of quantum field theories (whatever that might mean), with or without regulator. This is rather vague, but we argue below that we can ignore almost all the theories in $\mathcal{A}\setminus\mathcal{A}_{\text{reg}}$, as only a small fraction are \emph{physically relevant}. -Let's now construct a proper quantum field theory, i.e., a theory without cutoff. To do this suppose that for a particular given theory $(\mathcal{A}_\Lambda, \mathcal{E}_{t,\Lambda}, \omega_\Lambda)$ with observables $\mathcal{A}_\Lambda \subset \mathcal{A}_{\text{reg}}$ and cutoff $\Lambda$ we can always find a \emph{physically equivalent} theory $(\mathcal{A}_{\Lambda'}, \mathcal{E}_{t,\Lambda'}, \omega_{\Lambda'})$ with $\mathcal{A}_{\Lambda'} \subset \mathcal{A}_{\text{reg}}$ having a larger cutoff $\Lambda' > \Lambda$. If we can always do this then there is nothing stopping us sending the cutoff $\Lambda'\rightarrow \infty$ and calling the result a quantum field theory proper. Let's try and make this more concrete: what does it mean for a theory to have a larger cutoff than another theory? One clean operational interpretation is that all the \emph{effects} of our original theory can be found in the space of effects for the new theory with a larger cutoff. Thus, corresponding to the operation of changing cutoff from $\Lambda$ to $\Lambda' > \Lambda$, there must be a map +Let's now construct a proper quantum field theory, i.e., a theory without cutoff. To do this suppose that for a particular given theory $(\mathcal{A}_\Lambda, \mathcal{E}_{t,\Lambda}, \omega_\Lambda)$ with observables $\mathcal{A}_\Lambda \subset \mathcal{A}_{\text{reg}}$ and cutoff $\Lambda$ we can always find a \emph{physically equivalent} theory $(\mathcal{A}_{\Lambda'}, \mathcal{E}_{t,\Lambda'}, \omega_{\Lambda'})$ with $\mathcal{A}_{\Lambda'} \subset \mathcal{A}_{\text{reg}}$ having a larger cutoff $\Lambda' > \Lambda$. If we can always do this then there is nothing stopping us sending the cutoff $\Lambda'\rightarrow \infty$ and calling the result a quantum field theory proper. + +What does it mean for a theory to have a larger cutoff than another theory? One clean operational interpretation is that all the \emph{effects} of our original theory can be found in the space of effects for the new theory with a larger cutoff. Thus, corresponding to the operation of changing cutoff from $\Lambda$ to $\Lambda' > \Lambda$, there must be a map \begin{equation} \mathcal{F}_{\Lambda,\Lambda'}: \mathcal{A}_\Lambda \rightarrow \mathcal{A}_{\Lambda'} \end{equation} @@ -242,9 +242,9 @@ \section{The Wilsonian formulation of effective quantum field theory}\label{sec: \mathcal{A}_{\Lambda} \ar[rr]^{\mathcal{F}_{\Lambda,\Lambda'}} \ar[dr]_{f_\Lambda}\hole && \mathcal{A}_{\Lambda'} \ar[dl]^{f_\Lambda'}\\ -& \mathcal{A} } +& \mathcal{A}_{\text{reg}} } \end{equation} -It is usually assumed that that this flow on the infinite-dimensional space $\mathcal{A}$ is generated by a \emph{vector field}. A very special role is played by the \emph{fixed points} of this flow, as they correspond to genuine cutoff-free quantum field theories, i.e., theories of continuously many degrees of freedom. It is not at all obvious if $\mathcal{A}_{\text{reg}}$ contains any fixed points. This is an important observation, and it leads us closer to a resonable definition of $\mathcal{A}$ as the original space $\mathcal{A}_{\text{reg}}$ with missing points adjoined. Again the analogy with $\mathbb{Q}$ is helpful here: fixed points of well-defined maps on $\mathbb{Q}$ can easily fail to be in $\mathbb{Q}$, for example, consider $f(x) = x^2-1$: the fixed points of this map are $x_{\pm} = \frac{1\pm\sqrt{5}}{2}$. Thus we can tentatively think of $\mathcal{A}$ as the space of regulated theories with possibly missing fixed points of $\mathcal{F}_{\Lambda,\Lambda'}$ adjoined. It turns out that this picture is pretty close to the actual definition. Conformal field theories, being scale invariant, are then precisely fixed points. +It is usually assumed that that this flow on the infinite-dimensional space $\mathcal{A}_{\text{reg}}$ is generated by a \emph{vector field}, called the \emph{beta function}. A very special role is played by the \emph{fixed points} of this flow, as they correspond to genuine cutoff-free quantum field theories, i.e., theories of continuously many degrees of freedom. But does $\mathcal{A}_{\text{reg}}$ contain any such fixed points? This is an important question which leads us closer to a resonable definition of $\mathcal{A}$ as the original space $\mathcal{A}_{\text{reg}}$ with missing points adjoined. Again the analogy with $\mathbb{Q}$ is helpful here: fixed points of well-defined maps on $\mathbb{Q}$ can easily fail to be in $\mathbb{Q}$, for example, consider $f(x) = x^2-1$: the fixed points of this map are $x_{\pm} = \frac{1\pm\sqrt{5}}{2}$. Thus we can tentatively think of $\mathcal{A}$ as the space of regulated theories with possibly missing fixed points of $\mathcal{F}_{\Lambda,\Lambda'}$ adjoined. It turns out that this picture is pretty close to the actual definition. Conformal field theories, being scale invariant, are then precisely fixed points. It is standard to parametrise QFTs by \emph{local lagrangians}, which basically sort of amounts to a choice of a coordinate system for our mythical $\mathcal{A}$. Doing things this way tends to incentivise the conflation of both states and effects into one object via the \emph{path integral}. The space $\mathcal{M}$ of lagrangians is an infinite-dimensional linear manifold with a coordinate for each local term you can add, e.g., \begin{equation} @@ -254,9 +254,7 @@ \section{The Wilsonian formulation of effective quantum field theory}\label{sec: When we do things this way the maps $f_\Lambda$ and $\mathcal{F}_{\Lambda,\Lambda'}$ become \emph{diffeomorphisms} and the requirement that, after increasing the cutoff from $\Lambda$ to $\Lambda'$, the $n$-point correlation functions of large-scale low-energy observables remain invariant generates a deeply nontrivial \emph{renormalisation group} (RG) flow on the infinite dimensional manifold $\mathcal{M}$. -The point of view taken in this paper is to describe how to apply the Wilsonian view to different, indeed arbitrary, ways of parametrising QFTs. A key step is to first separating out states and observables into separate categories. - -As a crucial example, we'll show how to exploit \emph{tensor network states} to parametrise QFTs by implementing the continuum limit in the Wilsonian view directly on quantum states rather than lagrangians. +The point of view taken in this paper is to describe how to apply the Wilsonian view to different, indeed arbitrary, ways of parametrising QFTs. A key step is to first separating out states and observables into separate categories. As a crucial example, we'll show how to exploit \emph{tensor network states} to parametrise QFTs by implementing the continuum limit in the Wilsonian view directly on quantum states rather than lagrangians. \section{Effective quantum field states and the Wilsonian formulation}\label{sec:effectiveqftstates} diff --git a/What-is-a-quantum-field-state.bib b/What-is-a-quantum-field-state.bib index bd39feb..0e6f8ce 100644 --- a/What-is-a-quantum-field-state.bib +++ b/What-is-a-quantum-field-state.bib @@ -1,3 +1,10 @@ +@unpublished{pastawski_holographic_2015, + title = {Holographic quantum error-correcting codes: {Toy} models for the bulk/boundary correspondence}, + note = {arXiv:1503.06237}, + author = {Pastawski, Fernando and Yoshida, Beni and Harlow, Daniel and Preskill, John}, + year = {2015}, +} + @book{taylor_scattering_2006, address = {Mineola, N.Y.}, edition = {1st},