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Journal of Physics A: Mathematical and Theoretical

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Journal of Physics A: Mathematical and Theoretical is a major journal of theoretical physics reporting research on the mathematical structures that describe fundamental processes of the physical world and on the analytical, computational and numerical methods for exploring these structures.

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Ginestra Bianconi 2024 J. Phys. A: Math. Theor. 57 365002

We propose a theory for coupling matter fields with discrete geometry on higher-order networks, i.e. cell complexes. The key idea of the approach is to associate to a higher-order network the quantum entropy of its metric. Specifically we propose an action having two contributions. The first contribution is proportional to the logarithm of the volume associated to the higher-order network by the metric. In the vacuum this contribution determines the entropy of the geometry. The second contribution is the quantum relative entropy between the metric of the higher-order network and the metric induced by the matter and gauge fields. The induced metric is defined in terms of the topological spinors and the discrete Dirac operators. The topological spinors, defined on nodes, edges and higher-dimensional cells, encode for the matter fields. The discrete Dirac operators act on topological spinors, and depend on the metric of the higher-order network as well as on the gauge fields via a discrete version of the minimal substitution. We derive the coupled dynamical equations for the metric, the matter and the gauge fields, providing an information theory principle to obtain the field theory equations in discrete curved space.

Géza Tóth and Iagoba Apellaniz 2014 J. Phys. A: Math. Theor. 47 424006

We summarize important recent advances in quantum metrology, in connection to experiments in cold gases, trapped cold atoms and photons. First we review simple metrological setups, such as quantum metrology with spin squeezed states, with Greenberger–Horne–Zeilinger states, Dicke states and singlet states. We calculate the highest precision achievable in these schemes. Then, we present the fundamental notions of quantum metrology, such as shot-noise scaling, Heisenberg scaling, the quantum Fisher information and the Cramér–Rao bound. Using these, we demonstrate that entanglement is needed to surpass the shot-noise scaling in very general metrological tasks with a linear interferometer. We discuss some applications of the quantum Fisher information, such as how it can be used to obtain a criterion for a quantum state to be a macroscopic superposition. We show how it is related to the speed of a quantum evolution, and how it appears in the theory of the quantum Zeno effect. Finally, we explain how uncorrelated noise limits the highest achievable precision in very general metrological tasks.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

Giuseppe Gaeta and Epifanio G Virga 2023 J. Phys. A: Math. Theor. 56 363001

In its most restrictive definition, an octupolar tensor is a fully symmetric traceless third-rank tensor in three space dimensions. So great a body of works have been devoted to this specific class of tensors and their physical applications that a review would perhaps be welcome by a number of students. Here, we endeavour to place octupolar tensors into a broader perspective, considering non-vanishing traces and non-fully symmetric tensors as well. A number of general concepts are recalled and applied to either octupolar and higher-rank tensors. As a tool to navigate the diversity of scenarios we envision, we introduce the octupolar potential , a scalar-valued function which can easily be given an instructive geometrical representation. Physical applications are plenty; those to liquid crystal science play a major role here, as they were the original motivation for our interest in the topic of this review.

Michael (Misha) Chertkov 2024 J. Phys. A: Math. Theor. 57 333001

The paper reflects on the future role of artificial intelligence (AI) in scientific research, with a special focus on turbulence studies, and examines the evolution of AI, particularly through Diffusion Models rooted in non-equilibrium statistical mechanics. It underscores the significant impact of AI on advancing reduced, Lagrangian models of turbulence through innovative use of Deep Neural Networks. Additionally, the paper reviews various other AI applications in turbulence research and outlines potential challenges and opportunities in the concurrent advancement of AI and statistical hydrodynamics. This discussion sets the stage for a future where AI and turbulence research are intricately intertwined, leading to more profound insights and advancements in both fields.

Luca Angelani 2023 J. Phys. A: Math. Theor. 56 455003

The motion of run-and-tumble particles in one-dimensional finite domains are analyzed in the presence of generic boundary conditions. These describe accumulation at walls, where particles can either be absorbed at a given rate, or tumble, with a rate that may be, in general, different from that in the bulk. This formulation allows us to treat in a unified way very different boundary conditions (fully and partially absorbing/reflecting, sticky, sticky-reactive and sticky-absorbing boundaries) which can be recovered as appropriate limits of the general case. We report the general expression of the mean exit time, valid for generic boundaries, discussing many case studies, from equal boundaries to more interesting cases of different boundary conditions at the two ends of the domain, resulting in nontrivial expressions of mean exit times.

Zvi Bern et al 2024 J. Phys. A: Math. Theor. 57 333002

This review describes the duality between color and kinematics and its applications, with the aim of gaining a deeper understanding of the perturbative structure of gauge and gravity theories. We emphasize, in particular, applications to loop-level calculations, the broad web of theories linked by the duality and the associated double-copy structure, and the issue of extending the duality and double copy beyond scattering amplitudes. The review is aimed at doctoral students and junior researchers both inside and outside the field of amplitudes and is accompanied by various exercises.

Farhang Loran and Ali Mostafazadeh 2024 J. Phys. A: Math. Theor. 57 335205

Jing Liu et al 2020 J. Phys. A: Math. Theor. 53 023001

Quantum Fisher information matrix (QFIM) is a core concept in theoretical quantum metrology due to the significant importance of quantum Cramér–Rao bound in quantum parameter estimation. However, studies in recent years have revealed wide connections between QFIM and other aspects of quantum mechanics, including quantum thermodynamics, quantum phase transition, entanglement witness, quantum speed limit and non-Markovianity. These connections indicate that QFIM is more than a concept in quantum metrology, but rather a fundamental quantity in quantum mechanics. In this paper, we summarize the properties and existing calculation techniques of QFIM for various cases, and review the development of QFIM in some aspects of quantum mechanics apart from quantum metrology. On the other hand, as the main application of QFIM, the second part of this paper reviews the quantum multiparameter Cramér–Rao bound, its attainability condition and the associated optimal measurements. Moreover, recent developments in a few typical scenarios of quantum multiparameter estimation and the quantum advantages are also thoroughly discussed in this part.

Francisco J Sevilla et al 2024 J. Phys. A: Math. Theor. 57 335004

Katarzyna Siudzińska 2024 J. Phys. A: Math. Theor. 57 335302

Informationally overcomplete measurements find important applications in quantum tomography and quantum state estimation. The most popular are maximal sets of mutually unbiased bases, for which trace relations between measurement operators are well known. In this paper, we introduce a more general class of informationally overcomplete positive, operator-valued measure (POVMs) that are generated by equiangular tight frames of arbitrary rank. This class provides a generalization of equiangular measurements to non-projective POVMs, which include rescaled mutually unbiased measurements and bases. We provide a method of their construction, analyze their symmetry properties, and provide examples for highly symmetric cases. In particular, we find a wide class of generalized equiangular measurements that are conical two-designs, which allows us to derive the index of coincidence. Our results show benefits of considering a single informationally overcomplete measurement over informationally complete collections of POVMs.

Latest articles

Malik Mamode 2024 J. Phys. A: Math. Theor. 57 405201

The paper investigates the truncation error between the Green function and the lattice Green function (LGF) for the Laplacian operator defined on the 2-torus and its discretization on a regular square lattice. Extensions to the cylinder and the rectangular domain with free (or Neumann) boundary conditions are also proposed. In each of these instances, the Green function and its discrete analog are given in exact analytical closed-form allowing to infer accurate estimates as the lattice spacing tends to zero. As expected, it is shown that the continuum limit of the LGF coincides well with the Green function in every case. In particular, the issue of logarithmic singularity regularization of the Green function by the lattice discretization is addressed through two related application examples regarding the rectangular domain, and devoted to the computation of corner-to-corner resistance of an electrical conducting square and the mean first-passage time between the diagonally opposite vertices of a square for a standard Brownian motion, both derived considering the continuum limit.

Beatrice Nettuno et al 2024 J. Phys. A: Math. Theor. 57 405002

The general epidemic process (GEP), also known as susceptible-infected-recovered model , provides a minimal model of how an epidemic spreads within a population of susceptible individuals who acquire permanent immunization upon recovery. This model exhibits a second-order absorbing state phase transition, commonly studied assuming immobile healthy individuals. We investigate the impact of mobility on the scaling properties of disease spreading near the extinction threshold by introducing two generalizations of GEP, where the mobility of susceptible and recovered individuals is examined independently. In both cases, including mobility violates GEP's rapidity reversal symmetry and alters the number of absorbing states. The critical dynamics of the models are analyzed through a perturbative renormalization group (RG) approach and large-scale stochastic simulations using a Gillespie algorithm. The RG analysis predicts both models to belong to the same novel universality class describing the critical dynamics of epidemic spreading when the infected individuals interact with a diffusive species and gain immunization upon recovery. At the associated RG fixed point, the immobile species decouples from the dynamics of the infected species, dominated by the coupling with the diffusive species. Numerical simulations in two dimensions affirm our RG results by identifying the same set of critical exponents for both models. Violation of the rapidity reversal symmetry is confirmed by breaking the associated hyperscaling relation. Our study underscores the significance of mobility in shaping population spreading dynamics near the extinction threshold.

Amos Chan and Andrea De Luca 2024 J. Phys. A: Math. Theor. 57 405001

M V Berry and Pragya Shukla 2024 J. Phys. A: Math. Theor. 57 405302

Classical curl forces are position-dependent Newtonian forces (accelerations) that are not the gradient of a scalar potential, and in general cannot be described by Hamiltonians. However, a special class of curl forces can be described by Hamiltonians, with the unusual feature that the kinetic energy is anisotropic in the momentum components. Therefore they can be quantised conventionally. We quantise the simplest such case: motion in the plane, with a curl force azimuthally directed and linear. As expected, the quantum propagator, and the way this drives Gaussian wavepackets, directly reflects the spiralling classical curl force dynamics. Two classes of stationary states—eigenfunctions of a continuous spectrum for the unbounded Hamiltonian—are described. They possess unusual singularities and an unfamiliar quantisation condition; their explanation requires asymptotics and unfamiliar singularities in the underlying families of classical trajectories. The analysis is supported and illustrated numerically.

Pieter W Claeys and Austen Lamacraft 2024 J. Phys. A: Math. Theor. 57 405301

Review articles

D B Milošević et al 2024 J. Phys. A: Math. Theor. 57 393001

Norbert Büttgen and Hans-Albrecht Krug von Nidda 2024 J. Phys. A: Math. Theor. 57 313001

Based on a previous review on magnetic resonance in quantum spin chains (Krug von Nidda et al 2010 Eur. Phys. J. Spec. Top. 180 161–89) we report on further development in this field with special focus on transition–metal oxides and halogenides consisting of quasi one–dimensional spin systems, where both intra–and inter–chain exchange interaction may give rise to frustration effects and higher–order anisotropic exchange contributions like the Dzyaloshinskii–Moriya interaction become decisive for the formation of the magnetic ground state. Selected examples show how NMR and ESR contribute valuable information on the magnetic phases and exchange interactions involved: LiCuVO 4 with competing nearest neighbour and next–nearest neighbour intra–chain exchange, LiCu 2 O 2 with complex zig–zag chains, and Cs 2 CuCl 4 where the chains form a triangular lattice with the inter–chain interaction weaker but of the same order of magnitude than the intra–chain interaction. The so called paper–chain compound Ba 3 Cu 3 In 4 O 12 , where each successive pair of CuO 4 plaquettes is rotated by 90° with respect to its predecessor along the c –direction like in a paper–chain, provides an interesting topology of frustrated intra–chain exchange interactions. Finally, a few dimer systems are considered.

Iddo Eliazar 2024 J. Phys. A: Math. Theor. 57 233002

Diffusion is a generic term for random motions whose positions become more and more diffuse with time. Diffusion is of major importance in numerous areas of science and engineering, and the research of diffusion is vast and profound. This paper is the first in a stochastic 'intro series' to the multidisciplinary field of diffusion. The paper sets off from a basic question: how to quantitatively measure diffusivity? Having answered the basic question, the paper carries on to a follow-up question regarding statistical behaviors of diffusion: what further knowledge can the diffusivity measure provide, and when can it do so? The answers to the follow-up question lead to an assortment of notions and topics including: persistence and anti-persistence; aging and anti-aging; short-range and long-range dependence; the Wiener–Khinchin theorem and its generalizations; spectral densities, white noise, and their generalizations; and colored noises. Observing diffusion from a macro level, the paper culminates with: the universal emergence of power-law diffusivity; the three universal diffusion regimes—one regular, and two anomalous; and the universal emergence of 1/f noise. The paper is entirely self-contained, and its prerequisites are undergraduate mathematics and statistics.

Featured articles

Tim Adamo and Sumer Jaitly 2020 J. Phys. A: Math. Theor. 53 055401

Keith Alexander et al 2020 J. Phys. A: Math. Theor. 53 045001

We probe the character of knotting in open, confined polymers, assigning knot types to open curves by identifying their projections as virtual knots. In this sense, virtual knots are transitional, lying in between classical knot types, which are useful to classify the ambiguous nature of knotting in open curves. Modelling confined polymers using both lattice walks and ideal chains, we find an ensemble of random, tangled open curves whose knotting is not dominated by any single knot type, a behaviour we call weakly knotted. We compare cubically confined lattice walks and spherically confined ideal chains, finding the weak knotting probability in both families is quite similar and growing with length, despite the overall knotting probability being quite different. In contrast, the probability of weak knotting in unconfined walks is small at all lengths investigated. For spherically confined ideal chains, weak knotting is strongly correlated with the degree of confinement but is almost entirely independent of length. For ideal chains confined to tubes and slits, weak knotting is correlated with an adjusted degree of confinement, again with length having negligible effect.

Yongchao Lü and Joseph A Minahan 2020 J. Phys. A: Math. Theor. 53 024001

Accepted manuscripts

Chichinin 

An extension of the vector algebra for irreducible spherical tensor operators (ITOs) is proposed, which involves coupling two ITOs into a third one ( V (k v ) = T (k t ) × U (k u )* ), with the additional condition that one of the operators is a complex conjugate and therefore is not an ITO. The correctness condition is considered for this extension. The new coupling rule treats the density matrices ρ (K l ) and all tensors related to light polarization ( J (K r ) or Τ (K) ) equally.
This approach has been applied to rewrite the fundamental statistical equilibrium equations and radiation transfer equations relevant to astrophysical polarimetry. The revised equations are more concise and eliminate the need for the nj symbols. Instead, they utilize vector and scalar products of ITOs. All rate coefficients in the equations consist of two independent factors, the reduced matrix element from dipole moments and the product (vector or scalar) of the tensors ρ (K l ) and J (K r ) (or Τ (K) ).

Garilli et al 

Coarse-grained models are widely used to explain the effective behavior of partially observable physical systems with hidden degrees of freedom. Reduction procedures in state space typically disrupt Markovianity and a fluctuation relation cannot be formulated. A recently developed framework of transition-based coarse-graining gave rise to a fluctuation relation for a single current, while all others are hidden. Here, we extend the treatment to an arbitrary number of observable currents. Crucial for the derivation are the concepts of mixed currents and their conjugated effective affinities, that can be inferred from the time series of observable transitions. We also discuss the connection to generating functions, transient behavior, and how our result recovers the fluctuation relation for a complete set of currents.

Eryganov et al 

In this paper, a novel quantization scheme for cooperative games is proposed. The circuit is inspired by the Eisert-Wilkens-Lewenstein protocol, which was modified to represent cooperation between players and extended to 3--qubit states. The framework of Clifford algebra is used to perform necessary computations. In particular, we use a direct analogy between Dirac formalism and Quantum Register Algebra to represent circuits. This analogy enables us to perform automated proofs of the circuit equivalence in a simple fashion. The expected value of the Shapley value concerning quantum probabilities is employed to distribute players' payoffs after the measurement. We study how entanglement, representing the level of pre-agreement between players, affects the final utility distribution. The paper also demonstrates how the Quantum Register Algebra and GAALOP software can automate all necessary calculations.

Viennot 

We study the fuzzy spaces (as special examples of noncommutative manifolds) with their quasicoherent states in order to find their pertinent metrics. We show that they are naturally endowed with two natural "quantum metrics" which are associated with quantum fluctuations of "paths". The first one provides the length the mean path whereas the second one provides the average length of the fluctuated paths. Onto the classical manifold associated with the quasicoherent state (manifold of the mean values of the coordinate observables in the state minimising their quantum uncertainties) these two metrics provides two minimising geodesic equations. Moreover, fuzzy spaces being not torsion free, we have also two different autoparallel geodesic equations associated with two different adiabatic regimes in the move of a probe onto the fuzzy space. We apply these mathematical results to quantum gravity in BFSS matrix models, and to the quantum information theory of a controlled qubit submitted to noises of a large quantum environment.

Barkan 

This work explores the manner in which classical phase space distribution functions converge to the microcanonical distribution. We first prove a theorem about the lack of convergence, then define a generalization of the coarse-graining procedure that leads to convergence. We prove that the time evolution of phase space distributions is an isometry for a broad class of statistical distance metrics, implying that ensembles do not get any closer to (or farther from) equilibrium, according to these metrics. This extends the known result that strong convergence of phase space distributions to the microcanonical distribution does not occur. However, it has long been known that weak convergence can occur, such that coarse-grained distributions---defined by partitioning phase space into a finite number of cells---converge pointwise to the microcanonical distribution. We define a generalization of coarse-graining that removes the need for partitioning phase space into cells. We prove that our generalized coarse-grained distribution converges pointwise to the microcanonical distribution if the dynamics are strong mixing. As an example, we study an ensemble of triangular billiard systems.

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Ivan Eryganov et al 2024 J. Phys. A: Math. Theor.

Andrea Cavagna et al 2024 J. Phys. A: Math. Theor.

Experiments on bird flocks and midge swarms reveal that these natural systems are well described by an active theory in which conservation laws play a crucial role. By building a symplectic structure that couples the particles' velocities to the generator of their internal rotations (spin), the Inertial Spin Model (ISM) reinstates a second-order temporal dynamics that captures many phenomenological traits of flocks and swarms. The reversible structure of the ISM predicts that the total spin is a constant of motion, the central conservation law responsible for all the novel dynamical features of the model. However, fluctuations and dissipation introduced in the original model to make it relax, violate the spin conservation law, so that the ISM aligns with the biophysical phenomenology only within finite-size regimes, beyond which the overdamped dynamics characteristic of the Vicsek model takes over. Here, we introduce a novel version of the ISM, in which the irreversible terms needed to relax the dynamics strictly respect the conservation of the spin. We perform a numerical investigation of the fully conservative model, exploring both the fixed-network case, which belongs to the equilibrium class of Model G, and the active case, characterized by self-propulsion of the agents and an out-of-equilibrium reshuffling of the underlying interaction network. Our simulations not only capture the correct spin wave phenomenology of the ordered phase, but they also yield dynamical critical exponents in the near-ordering phase that agree very well with the theoretical predictions.

Alessandro Barbini and Luca Giuggioli 2024 J. Phys. A: Math. Theor.

We examine the diffusive dynamics of a lattice random walk subject to resetting in a one-dimensional spatially heterogeneous environment composed of two media separated by an interface. At random times the walker may reset its position to the interface, but only when in the left medium. In addition the spatial heterogeneity results from having unequal diffusivities and biases in the two media. We construct the Master equation for the dynamics of the walker occupation probability in unbounded space, solve it exactly in terms of generating functions, and analyse the dynamics of the first and second moment. Making use of the closed form solution in the unbounded case, we build the analytic solution of the Master equation in finite and semi-infinite domains. By bounding the space on the right with a reflecting boundary we study the first-passage dynamics to a single fully absorbing target placed in the left medium away from the interface. As reset strongly increases the time to reach the target, we find that the first-passage dynamics enter the motion-limited regime even for relative small resetting probability. We also identify a surprising non-monotonic dependence of the first-passage probability mode as a function of the bias. By deriving an analytic expression for the mean first-passage time, we show when its value is independent of the diffusivity and bias in the left medium, uncovering another example of the so-called mean disorder indifference phenomenon.

Marco Benedetti and Enrico Ventura 2024 J. Phys. A: Math. Theor.

The beneficial role of noise-injection in learning is a consolidated concept in the field of artificial neural networks, suggesting that even biological systems might take advantage of similar mechanisms to optimize their performance. The training-with-noise algorithm proposed by Gardner and collaborators is an emblematic example of a noise-injection procedure in recurrent networks, which can be used to model biological neural systems. We show how adding structure to noisy training data can substantially improve the algorithm performance, allowing the network to approach perfect retrieval of the memories and wide basins of attraction, even in the scenario of maximal injected noise. We also prove that the so-called Hebbian Unlearning rule coincides with the training-with-noise algorithm when noise is maximal and data are stable fixed points of the network dynamics.

Eli Hawkins et al 2024 J. Phys. A: Math. Theor. 57 395205

Geometric quantization is a natural way to construct quantum models starting from classical data. In this work, we start from a symplectic vector space with an inner product and—using techniques of geometric quantization—construct the quantum algebra and equip it with a distinguished state. We compare our result with the construction due to Sorkin—which starts from the same input data—and show that our distinguished state coincides with the Sorkin-Johnson state. Sorkin's construction was originally applied to the free scalar field over a causal set (locally finite, partially ordered set). Our perspective suggests a natural generalization to less linear examples, such as an interacting field.

Thomas E Baker and Negar Seif 2024 J. Phys. A: Math. Theor. 57 395306

We consider a set of density matrices. All of which are written in the same orbital basis, but the orbital basis size is less than the total Hilbert space size. We ask how each density matrix is related to each of the others by establishing a norm between density matrices based on the truncation error in a partial trace for a small set of orbitals. We find that states with large energy differences must have large differences in their density matrices. Small energy differences are divided into two groups, one where two density matrices have small differences and another where they are very different, as is the case of symmetry. We extend these ideas to a bundle of matrix product states and show that bond dimension of the wavefunction ansatz for two states with large energy differences are larger. Meanwhile, low energy differences can have nearly the same bond dimensions for similar states.

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Journal information

• 2007-present Journal of Physics A: Mathematical and Theoretical doi: 10.1088/issn.1751-8121 Online ISSN: 1751-8121 Print ISSN: 1751-8113

Journal history

• 2007-present Journal of Physics A: Mathematical and Theoretical
• 1975-2006 Journal of Physics A: Mathematical and General
• 1973-1974 Journal of Physics A: Mathematical, Nuclear and General
• 1968-1972 Journal of Physics A: General Physics

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Quantum physics and Einstein’s theory of general relativity are the two solid pillars that underlie much of modern physics. Understanding how these two well-established theories are related remains a central open question in theoretical physics.  Over the last several decades, efforts in this direction have led to a broad range of new physical ideas and mathematical tools.  In recent years, string theory and quantum field theory have converged in the context of holography, which connects quantum gravity in certain space-times with corresponding (conformal) field theories on a lower-dimensional space-time. These developments and connections have deepened our understanding not only of quantum gravity, cosmology, and particle physics, but also of intermediate scale physics, such as condensed matter systems, the quark-gluon plasma, and disordered systems.  String theory has also led to new insights to problems in many areas of mathematics.

The interface of quantum physics and gravity is currently leading to exciting new areas of progress, and is expected to remain vibrant in the coming decade.  Researchers in the Center for Theoretical Physics (CTP) have been at the forefront of many of the developments in these directions.  CTP faculty members work on string theory foundations, the range of solutions of the theory, general relativity and quantum cosmology, problems relating quantum physics to black holes, and the application of holographic methods to strongly coupled field theories.  The group in the CTP has close connections to condensed matter physicists, astrophysicists, and mathematicians both at MIT and elsewhere.

In recent years a set of new developments has begun to draw unexpected connections between a number of problems relating aspects of gravity, black holes, quantum information, and condensed matter systems. It is becoming clear that quantum entanglement, quantum error correction, and computational complexity play a fundamental role in the emergence of spacetime geometry through holographic duality.  Moreover these tools have led to substantial progress on the famous black hole information problem, giving new avenues for searching for a resolution of the tension between the physics of black holes and quantum mechanics.  CTP faculty members Netta Engelhardt and Daniel Harlow have been at the vanguard of these developments, which also tie into the research activity of several other CTP faculty members, including Aram Harrow , whose primary research focus is on quantum information, and Hong Liu , whose research connects black holes and quantum many-body dynamics.

Holographic dualities give both a new perspective into quantum gravitational phenomena as encoded in quantum field theory, and a way to explore aspects of strongly coupled field theories using the gravitational dual. CTP faculty have played a pioneering role in several applications of holographic duality. Hong Liu and Krishna Rajagopal are at the forefront of efforts that use holography to find new insights into the physics of the quark-gluon plasma. Liu was among the first to point out possible connections between black hole physics and the strange metal phase of high temperature superconductors, and in recent years has been combining insights from effective field theories, holography, and condensed matter physics to address various issues concerning far-from-equilibrium systems including superfluid turbulence, entanglement growth, quantum chaos, thermalization, and a complete formulation of fluctuating hydrodynamics. Gravitational effective field theories play a key role in the interpretation of gravitational wave observations. Mikhail Ivanov works at the intersection of these fields with the aim of testing strong field gravity at a new precision frontier.

Even though we understand string theory better than we did in decades past, there is still no clear fundamental description of the theory that works in all situations, and the set of four-dimensional solutions, or string vacua, is still poorly understood.  The work of Washington Taylor and Barton Zwiebach combines physical understanding with modern mathematical methods to address these questions, and has led to new insights into how observed physics fits into the framework of string theory as well as the development of new mathematical results and ideas. Alan Guth ‘s foundational work on inflationary cosmology has led him to focus on basic questions about the physics of the multiverse that arises naturally in the context of the many string theory vacua, and which provides the only current natural explanation for the observed small but positive cosmological constant.

Symmetry has long been a guiding principle in the study of quantum field theory and gravity. Shu-Heng Shao ’s research focuses on generalizations of global symmetries in field theory and lattice systems. These new symmetries and their anomalies lead to various new dynamical constraints on (de)confinement, scattering amplitudes, renormalization group flows, and more. They also unify different conjectures in quantum gravity and holography. The microscopic lattice realizations of these new symmetries are naturally expressed through tensor networks, indicating an intriguing link to quantum information theory. 

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Is theoretical physics in crisis?

By: Harriet Jarlett

Do recent discoveries mean there’s nothing left? Find out what the future holds for theoretical physics in our final In Theory series installment

Over the past decade physicists have explored new corners of our world, and in doing so have answered some of the biggest questions of the past century.

When researchers discovered the Higgs boson in 2012, it was a huge moment of achievement. It showed theorists had been right to look towards the Standard Model for answers about our Universe. But then the particle acted just like the theorists said it would, it obeyed every rule they predicted. If it had acted just slightly differently it would have raised many questions about the theory, and our universe. Instead, it raised few questions and gave no new clues about to where to look next.

In other words, the theorists had done too good a job.

Before these discoveries, physicists were standing on the edge of a metaphorical flat Earth, suspecting it was round but not knowing for sure.  Finding both the Higgs boson, and evidence of gravitational waves has brought scientists closer than ever to understanding two of the great theories of our time – the Standard Model and the theory of relativity.

Now the future of theoretical physics is at a critical point – they proved their own theories, so what is there to do now?

In an earlier article in this series, we spoke about how experimental physicists and theoretical physicists must work together. Their symbiotic relationship – with theorists telling experimentalists where to look, and experimentalists asking theorists for explanations of unusual findings – is necessary, if we are to keep making discoveries.

Just four years ago, in 2012, physicists still held a genuine uncertainty about whether the lynchpin of the Standard Model, the Higgs boson existed at all. Now, there’s much less uncertainty.

“We are still in an uncertain period, previously we were uncertain as to how the Standard Model could be completed. Now we know it is pretty much complete so we can focus on the questions beyond it, dark matter, the future of the universe, the beginning of the universe, little things like that,” says John Ellis, a theoretical physicist from Kings College, London who began working at CERN since 1973.

"We are struggling to find clear indications that can point us in the right direction. Some people see in this state of crisis a source of frustration. I see a source of excitement because new ideas have always thrived in moments of crisis." - Gian Giudice, head of the Theory Department at CERN.

With the discovery of the Higgs, there’s been a shift in this relationship, with theoreticians not necessarily leading the way. Instead, experiments look for data to try and give more evidence to the already proposed theories, and if something new is thrown up theorists scramble to explain and make sense of it.

"It’s like when you go mushroom hunting," says Michelangelo Mangano, a theoretical physicist who works closely with experimental physicists. "You spend all your energy looking, and at the end of the day you may not find anything. Here it’s the same, there is a lot of wasted energy because it doesn’t lead to much, but by exploring all corners of the field occasionally you find a little gold nugget, a perfect mushroom."

At the end of last year, both the ATLAS and CMS experiments at CERN found their mushroom, an intriguing, albeit very small, bump in the data.

This little, unexpected bump could be the door to a whole host of new physics, because it could be a new particle. After the discovery of the Higgs most of the holes in the Standard Model had been sewn up, but many physicists were optimistic about finding new anomalies.

"What happens in the future largely depends on what the LHC finds in its second run," Ellis explains. "So if it turns out that there’s no other new physics and we’re focusing on understanding the Higgs boson better, that’s a different possible future for physics than if LHC Run 2 finds a new particle we need to understand."

While the bump is too small for physicists to announce it conclusively, there’s been hundreds of papers published by theoretical physicists as they leap to say what it might be.

“Taking unexplained data, trying to fit it to your ideas about the universe, revising your ideas once you get more data, and on and on until you have unravelled the story of the universe – that’s the spirit of theoretical physics,” expresses Giudice. 

But we’ll only know whether it’s something worthwhile with the start of the LHC this month, May 2016, when experimental physicists can start to take even more data and conclude what it is. 

Next generation of theory

This unusual period of quiet in the world of theoretical physics means students studying physics might be more likely to go into experimental physics, where the major discoveries are seen as happening more often, and where young physicists have a chance to be the first to a discovery. 

Speaking to the Summer Students at CERN, some of whom hope to become theoretical physicists, there is the feeling that this period  of uncertainty makes following theory a luxury, one that young physicists, who need to have original ideas and publish lots of papers to get ahead, can’t afford.

Camille Bonvin is working as a fellow in the Theory Department on cosmology to try and understand why the universe is accelerating. If gravity is described by Einstein’s theory of general relativity the expansion should be slowing, not accelerating, which means there’s something we don’t understand. Bonvin is trying to find out what that is. She thinks the best theories are simple, consistent and make sense, like general relativity. "Einstein is completely logical, and his theory makes sense. Sometimes you have the impression of taking a theory which already exists and adding one element, then another, then another, to try and make the data fit it better, but its not a fundamental theory, so for me its not extremely beautiful."(Image: Sophia Bennett/CERN)

Camille Bonvin, a young theoretical physicist at CERN hopes that the data bump is the key to new physics, because without new discoveries it’s hard to keep a younger generation interested: “If both the LHC and the upcoming cosmological surveys find no new physics, it will be difficult to motivate new theorists. If you don't know where to go or what to look for, it's hard to see in which direction your research should go and which ideas you should explore.”

The future's bright

Richard Feynman, one of the most famous theoretical physicists once joked, "Physics is like sex. Sure, it may give some practical results, but that's not why we do it."

And Gian Giudice agrees –while the field’s current uncertainty makes it more difficult for young people to make breakthroughs, it’s not the promise of glory that encourages people to follow the theory path, but just a simple passion in why our universe is the way it is.

“It must be difficult for the new generations of young researchers to enter theoretical physics now when it is not clear where different directions are leading to,” he says. “But it's much more interesting to play when you don't know what's going to happen, rather than when the rules of the game have already been settled.”

Giudice, who took on the role of leading the theory department in January 2016 is optimistic that the turbulence the field currently faces makes it one of the most exciting times to become a theoretical physicist.

“It has often been said that it is difficult to make predictions; especially about the future. It couldn't be more true today in particle physics. This is what makes the present so exciting. Looking back in the history of physics you'll see that moments of crisis and confusion were invariably followed by great revolutionary ideas. I hope it's about to happen again,” smiles Giudice. 

To learn more about the theory department, read the rest of our In Theory series .

Welcome to the Theory corridor

Why bother with theoretical physics, are theoreticians just football fanatics, why are theoreticians filled with wanderlust..., which came first….

• Theoretical Astrophysics and Cosmology

Calculating and modeling the physics of the cosmos. First objects in the universe, relativistic astrophysics, neutron stars, black holes, inflation, cosmic evolution and structure.

Current research in theoretical astrophysics and cosmology at Stanford explores a wide range of critical questions. Major topics include numerical simulations of the formation of structure from small scales (first stars) to large scales (dark matter structure), galaxy formation, black holes (evolution, jets, accretion disks and orbiting objects), neutron stars (pulsars, magnetars), particle acceleration (relativistic shocks, origin of cosmic rays), gravitational lensing, and the very early universe (inflation).

See the section on  Observational & Experimental Astrophysics and Cosmology  for a description of how scientists at Stanford are probing many of these questions with experiments and telescopes.

The  Kavli Institute for Particle Astrophysics and Cosmology , housed in the the Fred Kavli Building at  SLAC  and the Physics & Astrophysics Building and the Varian Physics Laboratories on campus, hosts much of the research in astrophysics and cosmology at Stanford and SLAC. Scientists meet through twice-weekly  Tea Talks , a weekly  Cosmology Seminar  and a weekly  Astrophysics Colloquium . There are also close ties with the Stanford Institute for Theoretical Physics ( SITP ) and the  SLAC Theory Group .

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Risa Wechsler

• Galaxy Formation & Cosmology Group

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Methods of Theoretical Physics

• First Online: 11 June 2017

Cite this chapter

• Christa Jungnickel 4 &
• Russell McCormmach 4  

Part of the book series: Archimedes ((ARIM,volume 48))

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We discussed methods of research in theoretical physics in a general way in the first chapter. In this chapter we look at specific examples of methods as they entered research: Boltzmann’s use of the molecular method in the mechanical theory of heat; Hertz’s use of the method of mathematical phenomenology in electrodynamics; Planck’s use of principles as a method in physical chemistry; and Helmholtz’s and Boltzmann’s use of the method of analogy in developing heat and electromagnetic theory. For theoretical physics as a field, teaching is as important as research, for without the regular renewal of trained researchers, the field would stagnate and eventually come to an end. In this chapter, we look at Neumann’s, Kirchhoff’s, and Helmholtz’s methods of presenting theoretical physics in lectures.

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The extraordinary professorship for mathematical physics was created in 1863. Austrian Ministry of State, Section for Culture and Education, and Ministry of Finance reports, 10 July 1863, Öster. STA, 5 Phil, Physik. Hans Schobesberger, “Die Geschichte des Physikalischen Institutes der Universität Graz in den Jahren von 1850–1890” (manuscript), 16, Graz UA; Ministry of Culture and Education to Dean of the Philosophical Faculty, 15 April 1869; report by the Ministry to the Emperor, 28 June 1869; all in Öster. STA, 5 Phil. Physik; Theodor Des Coudres, “Ludwig Boltzmann,” Verh. Sächs. Ges. Wiss. 85 (1906): 615–27, on 623; Woldemar Voigt, “Ludwig Boltzmann,” Gött. Nachr. ,1907, 69–82, on 72.

Ludwig Boltzmann, “Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten,” Sitzungsber. Wiener Akad . 58 (1868): 517–60, reprinted in Wiss. Abh ., vol. 1, 49–96.

Ludwig Boltzmann, “Analytischer Beweis des zweiten Hauptsatzes der mechanischen Wärmetheorie aus den Sätzen über das Gleichgewicht der lebendigen Kraft,” Sitzungsber. Wiener Akad. 63 (1871): 712–32; repr. in Wiss. Abh. , vol. 1, 288–308, on 308.

Boltzmann, “Weitere Studien”; Martin J. Klein, Paul Ehrenfest, The Making of a Theoretical Physicist . (Amsterdam and London: North-Holland, 1970), vol. 1, 100.

Boltzmann, “Über das Wärmegleichgewicht,” 254–55; Martin J. Klein, “Maxwell, His Demon, and the Second Law of Thermodynamics,” American Scientist 58 (1970): 84–97, on 92.

Boltzmann, “Weitere Studien,” 369–402, especially 393. The reasons for Boltzmann’s choice of the letter E , in this form of writing the second law, and of the sign reversal (the entropy increases while E decreases in the passage to equilibrium) together with Boltzmann’s 1872 paper in general are discussed in Thomas S. Kuhn, Black-Body Theory and the Quantum Discontinuity 1894-1912 (New York: Oxford University Press, 1978), 42–46, 269n13.

For a time this statement was known as “Boltzmann’s minimum theory,” then as “Boltzmann’s H -theorem” (Stephen G. Brush, The Kind of Motion We Call Heat: A History of the Kinetic Theory of Gases in the 19th Century , vol. 1, Physics and the Atomists [Amsterdam and New York: North-Holland, 1976], 238; see 235–38 for Brush’s discussion of Boltzmann’s 1872 paper).

Boltzmann, “Weitere Studien,” 345.

Klein, Ehrenfest , 102.

Ludwig Boltzmann, “Über das Wirkungsgesetz der Molekularkräfte,” Sitzungsber. Wiener Akad. 66 (1872): 213–219; repr. Wiss. Abh. , vol. 1, 309–15.

Boltzmann, “Weitere Studien,” 368.

Ludwig Boltzmann, “Bemerkungen über einige Probleme der mechanischen Wärmetheorie,” Sitzungsber. Wiener Akad. 75 (1877): 62–100; repr. Wiss. Abh. , vol. 2, 112–48, especially 116–22; Klein, Ehrenfest , 102–4. Loschmidt’s statement, which came to be called the “reversibility paradox,” had been discussed by William Thomson in 1874 (Brush, Motion We Call Heat , vol. 1, 238–39).

Ludwig Boltzmann, “Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht,” Sitzungsber. Wiener Akad. 76 (1877): 373–435; repr. Wiss. Abh. , vol. 2, 164–223, on 165–66.

Boltzmann, “Über die Beziehung,” 168, 175–76, 190–93. Boltzmann’s reasoning in his 1877 paper is analyzed in detail, for example, in René Dugas, La théorie physique au sens de Boltzmann et ses prolongements modernes (Neuchâtel-Suisse: Griffon, 1959), 192–99; Klein, Ehrenfest , 105–8; Kuhn, Black-Body Theory , 47–54; Salvo D’Agostino, A History of the Ideas of Theoretical Physics, Essays on Nineteenth and Twentieth Century Physics (Dordrecht: Kluwer Academic Publishers, 2000), 212–13.

Boltzmann, “Über die Beziehung,” 217–18, 223.

Max Planck, “Gedächtnissrede auf Heinrich Hertz,” Verh. phys. Ges . 13 (1894): 9–29; repr. in Physikalische Abhandlungen und Vorträge, 3 vols. (Braunschweig: F. Vieweg, 1958), vol. 3, 268–88, on 281–82.

Hertz abandoned his earlier approach to Maxwell’s theory, in 1884, which treated electrical waves in the air and the ether without a physical hypothesis about the ether. In its place, he adopted Helmholtz’s dielectric polarization, which was important for his course of experiments. He retained his belief that Maxwell’s theory was superior, but Helmholtz’s approach allowed him to prove it experimentally without presuming its truth (D’Agostino, History of the Ideas of Theoretical Physics , 138, 149, 177–78, 181–82; Olivier Darrigol, Electrodynamics from Ampère to Einstein [Oxford: Oxford University Press, 2000], 238).

Planck, “Hertz,” 282. In time, in agreement with Maxwell, Hertz recognized only one electric force, and he turned his attention from the electromagnetic effects of changing polarizations in material dielectrics to the free propagation of electric waves.

Hertz to his parents, 13 November and 23 December 1887, 1 January 1888, in Heinrich Hertz, Erinnerungen, Briefe, Tagebücher , ed. J. Hertz, 2nd rev. ed. M. Hertz and C. Süsskind (San Francisco: San Francisco Press, 1977), 236–48.

Heinrich Hertz, “Ueber die Einwirkung einer gradlinigen elektrischen Schwingung auf eine benachbarte Strombahn,” Ann. 34 (1888): 155–70, on 169.

Heinrich Hertz, “Ueber die Ausbreitungsgeschwindigkeit der elektrodynamischen Wirkungen,” Ann. 34 (1888): 551–69, on 568–69; “Ueber elektrodynamische Wellen im Luftraume und deren Reflexion,” Ann. 34 (1888): 610–23, on 610.

Heinrich Hertz, “Die Kräfte elektrischer Schwingungen behandelt nach der Maxwell’schen Theorie,” Ann . 36 (1888): 1–22, on 1.

Heinrich Hertz, “Ueber Strahlen elektrischer Kraft,” Ann. 36 (1889): 769–83, on 781.

Eugen Goldstein, “Aus vergangenen Tagen der Berliner Physikalischen Gesellschaft,” Naturwiss. 13 (1925): 39–45, on 44.

Helmholtz’s preface to Hertz, Gesammelte Werke , vol. 3, Die Prinzipien der Mechanik, in neuem Zusammenhange dargestellt , ed. P. Lenard (Leipzig, 1894); The Principles of Mechanics Presented in a New Form, ed. D. E. Jones and J. T. Walley (London, 1899; reprint New York: Dover, 1956).

D’Agostino, History of the Ideas of Theoretical Physics , 122–23.

Heinrich Hertz, Ueber die Beziehungen zwischen Licht und Elektricität (Bonn, 1889); in Ges. Werke , vol. 1, 339–54, on 339–40, 344, 352–53.

Heinrich Hertz, “On the Fundamental Equations of Electromagnetics for Bodies at Rest,” 1890, in Electric Waves, Being Researches on the Propagation of Electric Action with Finite Velocity through Space, trans. D. E. Jones (New York, 1893; repr. New York: Dover, 1962), 201.

Hertz, “Fundamental Equations … at Rest,” 201. These became a standard form of the equations of Maxwell’s theory; Hertz’s left-handed coordinate system determines the sign. Hertz’s introduction of these equations is analyzed in Tetu Hirosige, “Electrodynamics before the Theory of Relativity, 1890-1905,” Jap. Stud. Hist. Sci . 5 (1966), 1–49, on 2–6; and in P. M. Heimann, “Maxwell, Hertz and the Nature of Electricity,” Isis 62 (1970): 149–57.

Ludwig Boltzmann, “Über die Entwicklung der Methoden der theoretischen Physik in neuerer Zeit,” in Populäre Schriften (Leipzig: J. A. Barth, 1905), 198–227, on 221; Mach to Hertz, 25 September 1890, Ms. Coll., DM, 2976.

Hertz, Ueber die Beziehungen , 353–54.

Report by the Berlin U. Philosophical Faculty, 29 November 1888, DZA, Merseburg; quoted in Armin Hermann, Max Planck in Selbstzeugnissen und Bilddokumenten (Reinbek b. Hamburg: Rowohlt, 1973), 21–22.

Max Planck, “Ueber das Princip der Vermehrung der Entropie. Erste Abhandlung. Gesetze des Verlaufs von Reactionen, die nach constanten Gewichtsverhältnissen vor sich gehen,” Ann 30 (1887): 562–82; repr. in Phys. Abh. , vol. 1, 196–216, on 196–200; Max Born, “Max Karl Ernst Ludwig Planck 1858-1947,” Obituary Notices of Fellows of the Royal Society 6 (1948): 161–88; repr. in Ausgewählte Abhandlungen , ed. Akademie der Wissenschaften in Göttingen (Göttingen: Vandenhoeck und Ruprecht, 1963), vol. 2, 626–46, on 629.

Einstein referred to the third paper of the series: Max Planck, “Ueber das Princip der Vermehrung der Entropie. Dritte Abhandlung. Gesetze des Eintritts beliebiger thermodynamischer and chemischer Reactionen,” Ann . 32 (1887): 462–503; repr. in Phys. Abh. , vol.3, 232–73; Albert Einstein, “Max Planck als Forscher,” Naturwiss . 1 (1913): 1077–79, on 1077.

Max Planck, “Allgemeines zur neueren Entwicklung der Wärmetheorie,” Zs. f. phys. Chemie 8 (1891): 647–56; repr. Phys. Abh. , vol. 1, 372–81, quotations on 380–81.

Proposal of Planck as ordinary member of the Prussian Academy of Sciences, signed by Helmholtz, Kundt, and Bezold. Document Nr. 23, in Physiker über Physiker , ed. Christa Kirsten, and Hans-Günther Körber (Berlin: Akademie-Verlag, 1975.), 125–26. The proposal is undated; Planck’s election was on 11 June 1894.

Maxwell was the first to work with general cyclic systems, applying them to electromagnetism. They were applied to heat by Maxwell, Rankine, and Helmholtz (Ludwig Boltzmann, Vorlesungen über die Prinzipe der Mechanik , vol. 2 [Leipzig: J. A. Barth, 1904], 166). The properties of monocyclic systems and Helmholtz’s reasons for studying them together with his conclusions are discussed in Martin J. Klein, “Mechanical Explanation at the End of the Nineteenth Century,” Centaurus 17 (1792): 58–82, on 63–67; Leo Königsberger, “The Investigations of Hermann von Helmholtz on the Fundamental Principles of Mathematics and Mechanics,” Annual Report of the … Smithsonian Institution … to July , 1896 (1898): 93–124, on 120–23; and, the main source of our discussion, Helmholtz’s Prussian Academy papers on the subject and his recapitulation of them in his Berlin lectures on the theory of heat: “Studien zur Statik monocyklischer Systeme,” Sitzungsber. preuss. Akad. , 1884, 159–77, 311–18, 755–59, and Vorlesungen über theoretische Physik , vol. 6, Vorlesungen über die Theorie der Wärme , ed. Franz Richarz (Leipzig: J. A. Barth, 1903), 338–70. We acknowledge discussions with Stephen M. Winters, who has made a study of Helmholtz’s physics, including his monocyclic systems and least-action principle.

Ludwig Boltzmann, Vorlesungen über Maxwells Theorie der Elektricität und des Lichtes , vol. 2, Verhältniss zur Fernwirkungstheorie; specielle Fälle der Elektrostatik, stationären Strömung und Induction (Leipzig, 1893), 22, 50.

Arnold Sommerfeld, “Das Institut für Theoretische Physik,”in Die wissenschaftlichen Anstalten der Ludwig-Maximilians-Universität zu München, ed. Karl Alexander von Müller (Munich: R. Oldenbourg und Dr. C. Wolf, 1926), 290–91, on 290.

Ludwig Boltzmann, “Über die Methoden der theoretischen Physik” (1892), in Populäre Schriften , 1–10. Boltzmann approved of Maxwell’s expression “dynamical illustration” for mechanisms representing the electromagnetic field. He discussed this and related points in Ludwig Boltzmann, Vorlesungen über Maxwells Theorie der Elektricität und des Lichtes , vol. 1, Ableitung der Grundgleichungen für ruhende, homogene, isotrope Körper (Leipzig, 1891), 13, 35, and elsewhere in his lectures.

Ludwig Boltzmann, “Über ein Medium, dessen mechanische Eigenschaften auf die von Maxwell für den Electromagnetismus aufgestellten Gleichungen führen,” Sitzungsber. bay. Akad. 22 (1892): 279–301; repr. in Wiss. Abh. , vol. 3, 406–27.

Ludwig Boltzmann, “Ueber die neueren Theorien der Elektrizität und des Magnetismus,” Verh. Ges. deutsch. Naturf. u. Ärzte 65 (1893): 34–35; repr. in Wiss. Abh. , vol. 3, 502–3.

Hertz to Kőnig, 20 April 1889, Bonn (DM 3195).

In 1834–1839, 41 individuals attended Neumann’s private lectures, but only 18 attended his seminar. In 1840–1849, 56 individuals attended his private lectures, 11 his seminar. For 1850–1859, 52 attended his private lectures, 34 his seminar. After 1860, the seminar became stronger: in 1860–1869, 84 attended the private lectures, 61 his seminar.

Franz Neumann, Vorlesungen über mathematische Physik, gehalten an der Universität Königsberg .  Einleitung in die theoretische Physik , ed. Carl Pape (Leipzig, 1883), vii, 1, 4, 78, 110, 117. The editor used lectures he took notes on in the winter 1858–1859, Paul Volkmann, Franz Neumann. 11. September 1798, 23. Mai 1895 [Leipzig, 1896], 38). Woldemar Voigt, “Zur Erinnerung an F. E. Neumann, gestorben am 23. Mai 1895 zu Königsberg i/Pr.,” Gött. Nachr ., 1895, 248–65, on 256, 258; repr. “Gedächtnissrede auf Franz Neumann,” in Franz Neumanns Gesammelte Werke , edited by his students, 3 vols. (Leipzig: B. G. Teubner, 1906–28), vol. 1, 3–19; Review of Neumann’s Einleitung in die theoretische Physik in Die Fortschritte der Physik im Jahre 1883 39 (1883): 166–67.

Gustav Kirchhoff, Vorlesungen über mathematische Physik , vol. 1, Mechanik , 3rd ed. (Leipzig, 1883), quotation from the preface to the first edition in 1876. The volume is based on lectures Kirchhoff gave at Heidelberg just before moving to Berlin. The third edition is almost unchanged from the first. Ludwig Boltzmann, Gustav Robert Kirchhoff (Leipzig, 1888), 22. Robert Helmholtz, “A Memoir of Gustav Robert Kirchhoff,” trans. J. de Perott, in Annual Report of the … Smithsonian Institution … to July , 1889 , 1890, 527–40, on 531.

Kirchhoff’s preface to Vorlesungen über Mechanik .

Kirchhoff, Vorlesungen über Mechanik , chapter 1.

Gustav Kirchhoff, Vorlesungen über mathematische Physik , vol. 2,  Vorlesungen über mathematische Optik , ed. K. Hensel (Leipzig, 1891). This volume is based on lectures he gave at Berlin in 1876–77 and 1885–86.

Gustav Kirchhoff, Vorlesungen über Electricität und Magnetismus , ed. M. Planck (Leipzig, 1891).

Gustav Kirchhoff, Vorlesungen über mathematische Physik , vol. 4, Vorlesungen über die Theorie der Wärme, ed. Max Planck (Leipzig, 1894), 1–5. This volume is based on lectures he gave at Berlin in 1876, 1878, 1880, 1882, and 1884.

Ibid., 5, 10, 12, 57.

Ibid., 51, 60–61, 69, 97, 102, 134–36.

Kirchhoff, Theorie der Wärme , 2nd and 3rd lectures.

Kirchhoff, Electricität und Magnetismus , 11.

Boltzmann, Kirchhoff , 25.

Planck’s foreword to Gustav Kirchhoff, Vorlesungen über mathematische Physik, vol. 3, Vorlesungen über Elektrictät und Magnetismus , ed. M. Planck (Leipzig, 1891).

Hermann von Helmholtz, Vorlesungen über theoretische Physik, vol. 1, Einleitung zu den Vorlesungen Über theoretische Physik, ed. A. König and C. Runge (Leipzig: J. A. Barth, 1903), pt. 1, 1, 7, 10–11, 14–16. Helmholtz’s introductory lectures were given at Berlin in 1893.

Helmholtz, Einleitung , 21.

Hermann von Helmholtz, Vorlesungen über theoretische Physik, vol. 2, Vorlesungen über die Dynamik continuirlich verbreiteter Massen, ed. Otto Krigar-Menzel (Leipzig: J. A. Barth, 1902), 1–2. The volumes of Helmholtz's lectures on the mechanics of mass points and of continuously distributed masses were based on lectures he gave at Berlin in 1893–94.

Ibid., 2–3.

Ibid., 7–8.

Hermann von Helmholtz, Vorlesungen über theoretische Physik, vol 6, Vorlesungen über die Theorie der Wärme , ed. Franz Richarz (Leipzig: J. A. Barth, 1903), 256–58. This volume was compiled from his notebooks for the summer semester of 1890, from stenographic notes taken of his lectures in the summer semester of 1893, and from notes taken by the editor of the volume in the early 1880s.

Helmholtz, Dynamik discreter Masseenpunkte, 231.

Hermann von Helmholtz, Vorlesungen über theoretische Physik , vol. 5, Vorlesungen über die elektromagnetische Theorie des Lichtes , ed. Arthur König and Carl Runge (Hamburg and Leipzig, 1897), 14–16.

Hermann von Helmholtz, Vorlesungen über theoretische Physik , vol. 4, Vorlesungen über Elektrodynamik und Theorie des Magnetismus , ed. O. Krigar-Menzel and M. Laue (Leipzig: J. A. Barth, 1907). Acknowledged by Emil Bose, in Phys. Zs. 9 (1908): 141.

Wilhelm Wien, “Helmholtz als Physiker.” Naturwiss . 9 (1921): 694–99, on 697.

Max von Laue, “Über Hermann von Helmholtz,” in Forschen und Wirken. Festschrift zur 150-Jahr-Feier der Humboldt-Universität zu Berlin 1810-1960 , vol. 1 (Berlin: VEB Deutscher Verlag der Wissenschaften, 1960), 359–66, on 360. In Wilhelm Ostwald’s division of scientists into two temperamental types, “Classical” and “Romantic,” Helmholtz was classical.

This is not just a play on words. Lewis Pyenson suggests that Boltzmann’s and Mach’s use of the word “classical” to describe traditional mechanics around 1890 would have been recognized at the time as having an affinity with the word appearing in “classical” philology, referring to the languages of ancient cultures, greatly admired in educated circles. This appears in Pyenson’s discussion of neo-humanism and mathematics in German education in the late nineteenth century ( The Young Einstein: The Advent of Relativity [Bristol and Boston: Adam Hilger, 1985], 175).

Karl Böhm, “H. von Helmholtz, Einleitung zu den Vorlesungen über theoretische Physik ,” Phys. Zs. 5 (1904): 140–43.

Wien, “Helmholtz als Physiker,” 699.

The association of the classical-modern distinction with the Solvay Congress has been proposed by Richard Staley. He points out that if it is meant to imply that physicists before 1900 were working in classical physics and physicists after 1900 were working in modern physics, it is a myth. There is a loose connection with the same terms, “classical” and “modern,” appearing in other parts of culture, such as art, at the same time ( Einstein’s Generation: The Origins of the Relativity Revolution [Chicago and London: University of Chicago Press, 2008], 348–49, 353, 355, 422).

Peter M. Harman, Energy, Force, and Matter: The Conceptual Development of Nineteenth-Century Physics (Cambridge: Cambridge University Press, 1982), 9–10.

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• A PhD / DPhil degree (or comparable) with a background in theoretical physics, applied mathematics or related disciplines from a recognized university;
• Prior experience with nonequilibrium statistical physics and a strong interest in active matter physics;
• Ability and desire to work in an international team on inter-disciplinary research topics;
• Good command of English, which is the working language of the department. German is an asset but not required.

We are offering excellent working conditions in a highly interdisciplinary and stimulating research environment. Salary is in accordance with the German state public service salary scale (E13 TVöD-Bund) and the corresponding social benefits. Working hours are full time. We offer opportunities regarding work life balance as well as health promotion services. The postdoctoral appointment is for two years. The starting date is flexible.

The Max-Planck Society is committed to achieving the highest level of excellence and diversity. We encourage applications from women, especially in areas where they are underrepresented, which includes theoretical physics. Moreover, we are committed to providing suitable working environment for everyone including individuals with disabilities.

Your application

To apply, please follow this link with the reference no MPIDS-W081 and submit your CV, publication list, a statement of research interest, and contacts of two referees. In addition to the description of your proposed research, it should also briefly describe your past and current research interests and why you are interested in joining our department. Processing of applications will start after the deadline of 10th January 2025, and will stop when the positions are filled.  

For questions, please contact:

Prof. Dr. Ramin Golestanian Tel. +49 551 5176-100 [email protected]

Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Ramin Golestanian Am Faßberg 17 37077 Göttingen Germany

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