#feynmandiagram — Public Fediverse posts

The #paperOfTheDay is "Duality of O(N) and Sp(N) random tensor models: tensors with symmetries" from 2023.
In previous weeks I have often mentioned the "vector model", which is the field-theoretic limit of a random distribution on a lattice where the elementary quantities do not just have a scalar numerical value (such as a temperature), but are a N-dimensional vector. Tensor models are a generalization of these, the "field" quantity is a tensor, i.e. an array of numbers of higher dimension (a vector is a 1-tensor and a matrix is a 2-tensor in this sense). Still, N is the size of these objects, so a 3-dimensional tensor would be a N x N x N array of numbers.
What is interesting is that these models often depend on the parameter N in a smooth way. Physically it makes no sense to have a "vector with 1.5 components", but mathematically the dependence on N is some polynomial, and one can insert any value. In particular, one can let N be negative. From a perspective of #Feynmandiagram s , the N-dependent polynomial counts all ways to find closed circuits in the graph, and assigns a factor N to each of them. If N is negative, each circuit gets a negative sign, which reminds of the behaviour of fermions. In fact, it had long been known that the O(N) vector model for negative N is equivalent to a vector model with positive N, but based upon the symplectic group Sp(N). The present article proves the analogous duality, O(-N)= Sp(N), for more general tensor models. For the moment this is a #mathematics statement, but in the long run such dualities have often given rise to unexpected insights about the #physics of such theories. https://iopscience.iop.org/article/10.1088/1751-8121/ad0af4

The #paperOfTheDay is "Duality of O(N) and Sp(N) random tensor models: tensors with symmetries" from 2023.
In previous weeks I have often mentioned the "vector model", which is the field-theoretic limit of a random distribution on a lattice where the elementary quantities do not just have a scalar numerical value (such as a temperature), but are a N-dimensional vector. Tensor models are a generalization of these, the "field" quantity is a tensor, i.e. an array of numbers of higher dimension (a vector is a 1-tensor and a matrix is a 2-tensor in this sense). Still, N is the size of these objects, so a 3-dimensional tensor would be a N x N x N array of numbers.
What is interesting is that these models often depend on the parameter N in a smooth way. Physically it makes no sense to have a "vector with 1.5 components", but mathematically the dependence on N is some polynomial, and one can insert any value. In particular, one can let N be negative. From a perspective of #Feynmandiagram s , the N-dependent polynomial counts all ways to find closed circuits in the graph, and assigns a factor N to each of them. If N is negative, each circuit gets a negative sign, which reminds of the behaviour of fermions. In fact, it had long been known that the O(N) vector model for negative N is equivalent to a vector model with positive N, but based upon the symplectic group Sp(N). The present article proves the analogous duality, O(-N)= Sp(N), for more general tensor models. For the moment this is a #mathematics statement, but in the long run such dualities have often given rise to unexpected insights about the #physics of such theories. https://iopscience.iop.org/article/10.1088/1751-8121/ad0af4

The #paperOfTheDay is "Duality of O(N) and Sp(N) random tensor models: tensors with symmetries" from 2023.
In previous weeks I have often mentioned the "vector model", which is the field-theoretic limit of a random distribution on a lattice where the elementary quantities do not just have a scalar numerical value (such as a temperature), but are a N-dimensional vector. Tensor models are a generalization of these, the "field" quantity is a tensor, i.e. an array of numbers of higher dimension (a vector is a 1-tensor and a matrix is a 2-tensor in this sense). Still, N is the size of these objects, so a 3-dimensional tensor would be a N x N x N array of numbers.
What is interesting is that these models often depend on the parameter N in a smooth way. Physically it makes no sense to have a "vector with 1.5 components", but mathematically the dependence on N is some polynomial, and one can insert any value. In particular, one can let N be negative. From a perspective of #Feynmandiagram s , the N-dependent polynomial counts all ways to find closed circuits in the graph, and assigns a factor N to each of them. If N is negative, each circuit gets a negative sign, which reminds of the behaviour of fermions. In fact, it had long been known that the O(N) vector model for negative N is equivalent to a vector model with positive N, but based upon the symplectic group Sp(N). The present article proves the analogous duality, O(-N)= Sp(N), for more general tensor models. For the moment this is a #mathematics statement, but in the long run such dualities have often given rise to unexpected insights about the #physics of such theories. https://iopscience.iop.org/article/10.1088/1751-8121/ad0af4

The #paperOfTheDay is "Duality of O(N) and Sp(N) random tensor models: tensors with symmetries" from 2023.
In previous weeks I have often mentioned the "vector model", which is the field-theoretic limit of a random distribution on a lattice where the elementary quantities do not just have a scalar numerical value (such as a temperature), but are a N-dimensional vector. Tensor models are a generalization of these, the "field" quantity is a tensor, i.e. an array of numbers of higher dimension (a vector is a 1-tensor and a matrix is a 2-tensor in this sense). Still, N is the size of these objects, so a 3-dimensional tensor would be a N x N x N array of numbers.
What is interesting is that these models often depend on the parameter N in a smooth way. Physically it makes no sense to have a "vector with 1.5 components", but mathematically the dependence on N is some polynomial, and one can insert any value. In particular, one can let N be negative. From a perspective of #Feynmandiagram s , the N-dependent polynomial counts all ways to find closed circuits in the graph, and assigns a factor N to each of them. If N is negative, each circuit gets a negative sign, which reminds of the behaviour of fermions. In fact, it had long been known that the O(N) vector model for negative N is equivalent to a vector model with positive N, but based upon the symplectic group Sp(N). The present article proves the analogous duality, O(-N)= Sp(N), for more general tensor models. For the moment this is a #mathematics statement, but in the long run such dualities have often given rise to unexpected insights about the #physics of such theories. https://iopscience.iop.org/article/10.1088/1751-8121/ad0af4

The #paperOfTheDay is "Duality of O(N) and Sp(N) random tensor models: tensors with symmetries" from 2023.
In previous weeks I have often mentioned the "vector model", which is the field-theoretic limit of a random distribution on a lattice where the elementary quantities do not just have a scalar numerical value (such as a temperature), but are a N-dimensional vector. Tensor models are a generalization of these, the "field" quantity is a tensor, i.e. an array of numbers of higher dimension (a vector is a 1-tensor and a matrix is a 2-tensor in this sense). Still, N is the size of these objects, so a 3-dimensional tensor would be a N x N x N array of numbers.
What is interesting is that these models often depend on the parameter N in a smooth way. Physically it makes no sense to have a "vector with 1.5 components", but mathematically the dependence on N is some polynomial, and one can insert any value. In particular, one can let N be negative. From a perspective of #Feynmandiagram s , the N-dependent polynomial counts all ways to find closed circuits in the graph, and assigns a factor N to each of them. If N is negative, each circuit gets a negative sign, which reminds of the behaviour of fermions. In fact, it had long been known that the O(N) vector model for negative N is equivalent to a vector model with positive N, but based upon the symplectic group Sp(N). The present article proves the analogous duality, O(-N)= Sp(N), for more general tensor models. For the moment this is a #mathematics statement, but in the long run such dualities have often given rise to unexpected insights about the #physics of such theories. https://iopscience.iop.org/article/10.1088/1751-8121/ad0af4

#paperOfTheDay is "Comparison between large-order estimates and perturbation series in a scalar field theory with Gaussian propagator" from 1978.
The perturbation series for small coupling in #quantumFieldTheory is usually divergent (because the theory with negative coupling describes different #physics and is not "smoothly" connected with the case of positive coupling). To obtain finite physical predictions for the positive coupling case, a #resummation is needed. In the 1970s and 1980s, the computational techniques were for the first time sufficient to reach, say, order 4 in perturbation series computed with #FeynmanDiagram s, and the question arose how to actually do the resummation in practice.
The present paper studies a model quantum field theory, where the propagator is exp(-p^2). This is not physically accurate, but it allows to compute perturbation series to very high order (order 8 here) because the #FeynmanIntegral s are easier. Secondly, one can carry out an #instanton calculation to determine the leading large-order growth. This allows to study whether or not the series coefficients match the expected growth rate, and how to use various different methods of resummation. In particular, the instanton analysis predicts the location and algebraic exponent for the leading singularity in the Borel plane, and this can be used as an additional input: Either, compute the Borel transform directly with a shifted-gamma function to account for the singularity, or make an ansatz of the type (known singularity)x(Pade approximant) for the Borel transformed function. The numerical data shows that this gives substantially higher accuracy for the resummation. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.17.2144

#paperOfTheDay is "Comparison between large-order estimates and perturbation series in a scalar field theory with Gaussian propagator" from 1978.
The perturbation series for small coupling in #quantumFieldTheory is usually divergent (because the theory with negative coupling describes different #physics and is not "smoothly" connected with the case of positive coupling). To obtain finite physical predictions for the positive coupling case, a #resummation is needed. In the 1970s and 1980s, the computational techniques were for the first time sufficient to reach, say, order 4 in perturbation series computed with #FeynmanDiagram s, and the question arose how to actually do the resummation in practice.
The present paper studies a model quantum field theory, where the propagator is exp(-p^2). This is not physically accurate, but it allows to compute perturbation series to very high order (order 8 here) because the #FeynmanIntegral s are easier. Secondly, one can carry out an #instanton calculation to determine the leading large-order growth. This allows to study whether or not the series coefficients match the expected growth rate, and how to use various different methods of resummation. In particular, the instanton analysis predicts the location and algebraic exponent for the leading singularity in the Borel plane, and this can be used as an additional input: Either, compute the Borel transform directly with a shifted-gamma function to account for the singularity, or make an ansatz of the type (known singularity)x(Pade approximant) for the Borel transformed function. The numerical data shows that this gives substantially higher accuracy for the resummation. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.17.2144

#paperOfTheDay is "Comparison between large-order estimates and perturbation series in a scalar field theory with Gaussian propagator" from 1978.
The perturbation series for small coupling in #quantumFieldTheory is usually divergent (because the theory with negative coupling describes different #physics and is not "smoothly" connected with the case of positive coupling). To obtain finite physical predictions for the positive coupling case, a #resummation is needed. In the 1970s and 1980s, the computational techniques were for the first time sufficient to reach, say, order 4 in perturbation series computed with #FeynmanDiagram s, and the question arose how to actually do the resummation in practice.
The present paper studies a model quantum field theory, where the propagator is exp(-p^2). This is not physically accurate, but it allows to compute perturbation series to very high order (order 8 here) because the #FeynmanIntegral s are easier. Secondly, one can carry out an #instanton calculation to determine the leading large-order growth. This allows to study whether or not the series coefficients match the expected growth rate, and how to use various different methods of resummation. In particular, the instanton analysis predicts the location and algebraic exponent for the leading singularity in the Borel plane, and this can be used as an additional input: Either, compute the Borel transform directly with a shifted-gamma function to account for the singularity, or make an ansatz of the type (known singularity)x(Pade approximant) for the Borel transformed function. The numerical data shows that this gives substantially higher accuracy for the resummation. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.17.2144

#paperOfTheDay is "Comparison between large-order estimates and perturbation series in a scalar field theory with Gaussian propagator" from 1978.
The perturbation series for small coupling in #quantumFieldTheory is usually divergent (because the theory with negative coupling describes different #physics and is not "smoothly" connected with the case of positive coupling). To obtain finite physical predictions for the positive coupling case, a #resummation is needed. In the 1970s and 1980s, the computational techniques were for the first time sufficient to reach, say, order 4 in perturbation series computed with #FeynmanDiagram s, and the question arose how to actually do the resummation in practice.
The present paper studies a model quantum field theory, where the propagator is exp(-p^2). This is not physically accurate, but it allows to compute perturbation series to very high order (order 8 here) because the #FeynmanIntegral s are easier. Secondly, one can carry out an #instanton calculation to determine the leading large-order growth. This allows to study whether or not the series coefficients match the expected growth rate, and how to use various different methods of resummation. In particular, the instanton analysis predicts the location and algebraic exponent for the leading singularity in the Borel plane, and this can be used as an additional input: Either, compute the Borel transform directly with a shifted-gamma function to account for the singularity, or make an ansatz of the type (known singularity)x(Pade approximant) for the Borel transformed function. The numerical data shows that this gives substantially higher accuracy for the resummation. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.17.2144

#paperOfTheDay is "Comparison between large-order estimates and perturbation series in a scalar field theory with Gaussian propagator" from 1978.
The perturbation series for small coupling in #quantumFieldTheory is usually divergent (because the theory with negative coupling describes different #physics and is not "smoothly" connected with the case of positive coupling). To obtain finite physical predictions for the positive coupling case, a #resummation is needed. In the 1970s and 1980s, the computational techniques were for the first time sufficient to reach, say, order 4 in perturbation series computed with #FeynmanDiagram s, and the question arose how to actually do the resummation in practice.
The present paper studies a model quantum field theory, where the propagator is exp(-p^2). This is not physically accurate, but it allows to compute perturbation series to very high order (order 8 here) because the #FeynmanIntegral s are easier. Secondly, one can carry out an #instanton calculation to determine the leading large-order growth. This allows to study whether or not the series coefficients match the expected growth rate, and how to use various different methods of resummation. In particular, the instanton analysis predicts the location and algebraic exponent for the leading singularity in the Borel plane, and this can be used as an additional input: Either, compute the Borel transform directly with a shifted-gamma function to account for the singularity, or make an ansatz of the type (known singularity)x(Pade approximant) for the Borel transformed function. The numerical data shows that this gives substantially higher accuracy for the resummation. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.17.2144

#paperOfTheDay is "Effective five-wave Hamiltonian for surface water waves" from 1997. This article concerns (a certain idealization of) waves on the surface of deep water in one spacial dimension. These waves have a non-linear equation of motion, therefore they can "scatter" when they meet. This has nothing to do with #quantumFieldTheory , but still, the scattering amplitudes can be computed with #FeynmanDiagram s: The sum of all tree-level diagrams reproduces the non-quantum classical field theory. Such computation has the advantage that it is more structured, and ultimately simpler, than "manually" modifying and expanding the interaction Hamiltonian in superpositions of plane waves.
Since 1-dimensional water waves are not the most fashionable area of theoretical #physics right now, this work was maybe not widely known. However, it has resurfaced now as part of an ongoing effort trying to compute, or re-interpret, sums of tree-level Feynman diagrams (=non-quantum scattering amplitudes) in more geometric terms. While these endeavors in quantum field theory always have the shortcoming of leaving out the actual quantum part, for a classical field theory such as water waves, the sum of tree-level diagrams is a genuine physically relevant observable.
https://www.sciencedirect.com/science/article/pii/S0375960197002107?__cf_chl_f_tk=rgYKJNN8nI6263mmLkztzvasv.4l8IQQrTh0qPjse9Y-1783072682-1.0.1.1-BKQujlGYbuT1Onn9XcARc8fj7OS2.QuI8_CKS90K3dA

#paperOfTheDay is "Effective five-wave Hamiltonian for surface water waves" from 1997. This article concerns (a certain idealization of) waves on the surface of deep water in one spacial dimension. These waves have a non-linear equation of motion, therefore they can "scatter" when they meet. This has nothing to do with #quantumFieldTheory , but still, the scattering amplitudes can be computed with #FeynmanDiagram s: The sum of all tree-level diagrams reproduces the non-quantum classical field theory. Such computation has the advantage that it is more structured, and ultimately simpler, than "manually" modifying and expanding the interaction Hamiltonian in superpositions of plane waves.
Since 1-dimensional water waves are not the most fashionable area of theoretical #physics right now, this work was maybe not widely known. However, it has resurfaced now as part of an ongoing effort trying to compute, or re-interpret, sums of tree-level Feynman diagrams (=non-quantum scattering amplitudes) in more geometric terms. While these endeavors in quantum field theory always have the shortcoming of leaving out the actual quantum part, for a classical field theory such as water waves, the sum of tree-level diagrams is a genuine physically relevant observable.
https://www.sciencedirect.com/science/article/pii/S0375960197002107?__cf_chl_f_tk=rgYKJNN8nI6263mmLkztzvasv.4l8IQQrTh0qPjse9Y-1783072682-1.0.1.1-BKQujlGYbuT1Onn9XcARc8fj7OS2.QuI8_CKS90K3dA

#paperOfTheDay is "Effective five-wave Hamiltonian for surface water waves" from 1997. This article concerns (a certain idealization of) waves on the surface of deep water in one spacial dimension. These waves have a non-linear equation of motion, therefore they can "scatter" when they meet. This has nothing to do with #quantumFieldTheory , but still, the scattering amplitudes can be computed with #FeynmanDiagram s: The sum of all tree-level diagrams reproduces the non-quantum classical field theory. Such computation has the advantage that it is more structured, and ultimately simpler, than "manually" modifying and expanding the interaction Hamiltonian in superpositions of plane waves.
Since 1-dimensional water waves are not the most fashionable area of theoretical #physics right now, this work was maybe not widely known. However, it has resurfaced now as part of an ongoing effort trying to compute, or re-interpret, sums of tree-level Feynman diagrams (=non-quantum scattering amplitudes) in more geometric terms. While these endeavors in quantum field theory always have the shortcoming of leaving out the actual quantum part, for a classical field theory such as water waves, the sum of tree-level diagrams is a genuine physically relevant observable.
https://www.sciencedirect.com/science/article/pii/S0375960197002107?__cf_chl_f_tk=rgYKJNN8nI6263mmLkztzvasv.4l8IQQrTh0qPjse9Y-1783072682-1.0.1.1-BKQujlGYbuT1Onn9XcARc8fj7OS2.QuI8_CKS90K3dA

#paperOfTheDay is "Effective five-wave Hamiltonian for surface water waves" from 1997. This article concerns (a certain idealization of) waves on the surface of deep water in one spacial dimension. These waves have a non-linear equation of motion, therefore they can "scatter" when they meet. This has nothing to do with #quantumFieldTheory , but still, the scattering amplitudes can be computed with #FeynmanDiagram s: The sum of all tree-level diagrams reproduces the non-quantum classical field theory. Such computation has the advantage that it is more structured, and ultimately simpler, than "manually" modifying and expanding the interaction Hamiltonian in superpositions of plane waves.
Since 1-dimensional water waves are not the most fashionable area of theoretical #physics right now, this work was maybe not widely known. However, it has resurfaced now as part of an ongoing effort trying to compute, or re-interpret, sums of tree-level Feynman diagrams (=non-quantum scattering amplitudes) in more geometric terms. While these endeavors in quantum field theory always have the shortcoming of leaving out the actual quantum part, for a classical field theory such as water waves, the sum of tree-level diagrams is a genuine physically relevant observable.
https://www.sciencedirect.com/science/article/pii/S0375960197002107?__cf_chl_f_tk=rgYKJNN8nI6263mmLkztzvasv.4l8IQQrTh0qPjse9Y-1783072682-1.0.1.1-BKQujlGYbuT1Onn9XcARc8fj7OS2.QuI8_CKS90K3dA

#paperOfTheDay is "Effective five-wave Hamiltonian for surface water waves" from 1997. This article concerns (a certain idealization of) waves on the surface of deep water in one spacial dimension. These waves have a non-linear equation of motion, therefore they can "scatter" when they meet. This has nothing to do with #quantumFieldTheory , but still, the scattering amplitudes can be computed with #FeynmanDiagram s: The sum of all tree-level diagrams reproduces the non-quantum classical field theory. Such computation has the advantage that it is more structured, and ultimately simpler, than "manually" modifying and expanding the interaction Hamiltonian in superpositions of plane waves.
Since 1-dimensional water waves are not the most fashionable area of theoretical #physics right now, this work was maybe not widely known. However, it has resurfaced now as part of an ongoing effort trying to compute, or re-interpret, sums of tree-level Feynman diagrams (=non-quantum scattering amplitudes) in more geometric terms. While these endeavors in quantum field theory always have the shortcoming of leaving out the actual quantum part, for a classical field theory such as water waves, the sum of tree-level diagrams is a genuine physically relevant observable.
https://www.sciencedirect.com/science/article/pii/S0375960197002107?__cf_chl_f_tk=rgYKJNN8nI6263mmLkztzvasv.4l8IQQrTh0qPjse9Y-1783072682-1.0.1.1-BKQujlGYbuT1Onn9XcARc8fj7OS2.QuI8_CKS90K3dA

#paperOfTheDay for Friday is "Unitarity violation at the Wilson-Fisher fixed point in 4-epsilon dimensions" from 2016.
Statistical #physics and #quantumFieldTheory usually involve two parameters that physically only allow integer values: The dimension of space(time), and the dimensions of internal symmetry groups (such as the SU(3) in the standard model QCD, or O(N) symmetry in scalar fields). On the other hand, it is routine to formally assign non-integer values to them. Dimensional regularization sets D=4-epsilon, where epsilon is not assumed to be integer, and the #FeynmanDiagram s of O(N) symmetric theories are polynomials in N, hence allowing any value.
The present article points out that even a free, 1-component scalar field theory contains states with negative norm if one lets D be non-integer. The argument is surprisingly simple: Consider operators which are built from spacetime-derivatives d_mu acting on fields. In particular, we are interested in those operators which are antisymmetric in their indices. However, in D integer dimensions, there are D coordinate directions, and hence an operator with n>D derivatives can not be fully antisymmetric. Hence, the antisymmetric operators vanish when n>D for integer D. This does not hold for non-integer D, so that the operator actually has zeros at all the integer D. One can then see, by explicit calculation, that the 2-point functions of such operators (in the free theory!) flip sign at integer D, hence they are sometimes negative, hence the theory is not unitary.
This shows that the extension to non-integer D is very subtle; similar trouble exists for the Dirac matrices gamma_mu.
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.93.125025

#paperOfTheDay for Friday is "Unitarity violation at the Wilson-Fisher fixed point in 4-epsilon dimensions" from 2016.
Statistical #physics and #quantumFieldTheory usually involve two parameters that physically only allow integer values: The dimension of space(time), and the dimensions of internal symmetry groups (such as the SU(3) in the standard model QCD, or O(N) symmetry in scalar fields). On the other hand, it is routine to formally assign non-integer values to them. Dimensional regularization sets D=4-epsilon, where epsilon is not assumed to be integer, and the #FeynmanDiagram s of O(N) symmetric theories are polynomials in N, hence allowing any value.
The present article points out that even a free, 1-component scalar field theory contains states with negative norm if one lets D be non-integer. The argument is surprisingly simple: Consider operators which are built from spacetime-derivatives d_mu acting on fields. In particular, we are interested in those operators which are antisymmetric in their indices. However, in D integer dimensions, there are D coordinate directions, and hence an operator with n>D derivatives can not be fully antisymmetric. Hence, the antisymmetric operators vanish when n>D for integer D. This does not hold for non-integer D, so that the operator actually has zeros at all the integer D. One can then see, by explicit calculation, that the 2-point functions of such operators (in the free theory!) flip sign at integer D, hence they are sometimes negative, hence the theory is not unitary.
This shows that the extension to non-integer D is very subtle; similar trouble exists for the Dirac matrices gamma_mu.
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.93.125025

Yesterday's #paperOfTheDay was "Critical Equation of State from the Average Action" from 1996. This paper is one of the first applications of the Wetterich equation: A numerical solution of the local potential approximation for the vector model. The vector model is physically interesting in 3 dimensions.
In conventional perturbative #QuantumFieldTheory, one would probably start in 4-2epsilon dimensions, and compute a power series in epsilon with #FeynmanDiagram s, to then arrive at a 3-dimensional theory with epsilon=1/2. The Feynman diagrams have 4-valent vertices, which is thought of as a microscopic point-like "collision" between 4 "particles".
In the functional/statistical #physics perspective on field theory, the same situation is interpreted quite differently. One is in 3 dimensions throughout, and considers a system (e.g. lattice) where the constituents do not move. A phi^4-term is then a potential, i.e. every individual particle oscillates in its own local quartic potential. Additionally, the particles are coupled to neighbours. This is the "microscopic" theory, represented by the classical action. At longer distances, one effectively merges many of the lattice sites, and the so-obtained average quantities have all sorts of complicated interactions. The present paper uses the "local potential" approximation, which says that the only coupling to neighbours is still an elastic next-neighbour interaction (i.e. standard kinetic term p^2 in momentum space), and all that changes is that the particles are now in a more complicated potential. In particular, this might have non-trivial minima, which reflects a broken symmetry. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.77.873

Yesterday's #paperOfTheDay was "Critical Equation of State from the Average Action" from 1996. This paper is one of the first applications of the Wetterich equation: A numerical solution of the local potential approximation for the vector model. The vector model is physically interesting in 3 dimensions.
In conventional perturbative #QuantumFieldTheory, one would probably start in 4-2epsilon dimensions, and compute a power series in epsilon with #FeynmanDiagram s, to then arrive at a 3-dimensional theory with epsilon=1/2. The Feynman diagrams have 4-valent vertices, which is thought of as a microscopic point-like "collision" between 4 "particles".
In the functional/statistical #physics perspective on field theory, the same situation is interpreted quite differently. One is in 3 dimensions throughout, and considers a system (e.g. lattice) where the constituents do not move. A phi^4-term is then a potential, i.e. every individual particle oscillates in its own local quartic potential. Additionally, the particles are coupled to neighbours. This is the "microscopic" theory, represented by the classical action. At longer distances, one effectively merges many of the lattice sites, and the so-obtained average quantities have all sorts of complicated interactions. The present paper uses the "local potential" approximation, which says that the only coupling to neighbours is still an elastic next-neighbour interaction (i.e. standard kinetic term p^2 in momentum space), and all that changes is that the particles are now in a more complicated potential. In particular, this might have non-trivial minima, which reflects a broken symmetry. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.77.873

#paperOfTheDay "Non-Wilsonian ultraviolet completion via transseries" from 2021. A #quantumFieldTheory with marginally renormalizable coupling, such as the standard model of particle #physics usually leads to a power series solution that is divergent in two different ways: The number of #FeynmanDiagram s grows too fast, and the renormalized value of individual diagrams grows too fast. The latter is called #renormalon , and it can also be found in many other frameworks of QFT.
The present paper uses an analysis based on the renormalization group equation, truncated to low-order terms, to argue that the presence of the renormalon implies an ambiguity in the resummation. The technical machinery for this is called "resurgence", the basic mechanism is really intuitive: The Borel resummation is an integral along the positive real line, the renormalon is an algebraic singularity on that line, hence there is an ambiguity of which side (and how often, etc) one wants to pass the singularity. The paper arrives at two possible, mutually exclusive, interpretations of this finding.
I find these considerations really exciting, and they are closely related to my own work. However, I think it is fair to say that the many papers that have been written about renormalon chain resummation often raise more new questions than they answer, and at least to me the "big picture" of how this is supposed to work beyond leading-order is largely unclear. https://doi.org/10.1142/S0217751X21500160

#paperOfTheDay "Non-Wilsonian ultraviolet completion via transseries" from 2021. A #quantumFieldTheory with marginally renormalizable coupling, such as the standard model of particle #physics usually leads to a power series solution that is divergent in two different ways: The number of #FeynmanDiagram s grows too fast, and the renormalized value of individual diagrams grows too fast. The latter is called #renormalon , and it can also be found in many other frameworks of QFT.
The present paper uses an analysis based on the renormalization group equation, truncated to low-order terms, to argue that the presence of the renormalon implies an ambiguity in the resummation. The technical machinery for this is called "resurgence", the basic mechanism is really intuitive: The Borel resummation is an integral along the positive real line, the renormalon is an algebraic singularity on that line, hence there is an ambiguity of which side (and how often, etc) one wants to pass the singularity. The paper arrives at two possible, mutually exclusive, interpretations of this finding.
I find these considerations really exciting, and they are closely related to my own work. However, I think it is fair to say that the many papers that have been written about renormalon chain resummation often raise more new questions than they answer, and at least to me the "big picture" of how this is supposed to work beyond leading-order is largely unclear. https://doi.org/10.1142/S0217751X21500160

#paperOfTheDay is "Effective field equations for expectation values" from 1986. Scattering in #quantumFieldTheory is usually defined as a time evolution from some infinite-past to infinite-future state, both of which are plane waves (=#particles on straight trajectories). In this setup, a natural definition for an "expectation value" of a field variable is the expectation between these states, i.e. <in | phi(x) | out> . The present paper introduces another class of expectation values, of the form <in | phi(x) | in>, and derive various equations for them. These,equations are different, but structurally equivalent, to the usual ones (e.g. if one uses perturbation theory, there still are the usual #FeynmanDiagram s, but one uses a retarded propagator in place of a Feynman propagator). The crucial difference is the interpretation of these quantities: The paper contains a nice example for a system where the |in> and |out> states have different plane wave states (i.e. a harmonic oscillator where the spring constant changes over time). In that case, it can be hard to interpret the conventional expectation values, whereas the new ones always refer to the |in> plane wave basis. Also, the conventional setup requires boundary conditions at the past and the future to compute the evaluation of the mean field, while the new setup can compute time evolution from just initial conditions in the past, which is more natural in certain quasi-classical setups. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.33.444

#paperOfTheDay is "Effective field equations for expectation values" from 1986. Scattering in #quantumFieldTheory is usually defined as a time evolution from some infinite-past to infinite-future state, both of which are plane waves (=#particles on straight trajectories). In this setup, a natural definition for an "expectation value" of a field variable is the expectation between these states, i.e. <in | phi(x) | out> . The present paper introduces another class of expectation values, of the form <in | phi(x) | in>, and derive various equations for them. These,equations are different, but structurally equivalent, to the usual ones (e.g. if one uses perturbation theory, there still are the usual #FeynmanDiagram s, but one uses a retarded propagator in place of a Feynman propagator). The crucial difference is the interpretation of these quantities: The paper contains a nice example for a system where the |in> and |out> states have different plane wave states (i.e. a harmonic oscillator where the spring constant changes over time). In that case, it can be hard to interpret the conventional expectation values, whereas the new ones always refer to the |in> plane wave basis. Also, the conventional setup requires boundary conditions at the past and the future to compute the evaluation of the mean field, while the new setup can compute time evolution from just initial conditions in the past, which is more natural in certain quasi-classical setups. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.33.444

Many computations in #quantum field theory are "perturbative", that means that one computes a series expansion in powers of some parameter, called g. The usual intuition is that the parameter is small, so that higher powers of the parameter are negligible. However, in most cases, these series are divergent so that their sum is infinite for every non-zero value of the parameter g. This sometimes leads to confusion. The solution is to realize that there exist perfectly fine functions where a power series expansion is divergent. Not every function has a convergent series expansion. In the context of quantum field theory, one can see this explicitly in the case of a zero-dimensional theory. In that case, every #FeynmanDiagram is assigned the value 1 (which is a huge simplification). One can then solve the entire perturbation series. It is divergent, but it can equivalently be represented as a convergent integral. The conclusion is that quantum field theory delivers a finite, smooth function, but its perturbation series diverges since that function is imaginary for certain parameter ranges (which has physical interpretation as decay of particles). The mathematics is fine, the problem is that physics education is very much focused on polynomials and Taylor series, which leads to the wrong expectation that every physically sensible function should have a convergent Taylor series expansion. I recently gave a talk about this at PI, slides are available on my homepage https://paulbalduf.com/research/vector-asymptotics/
#physics #math

Many computations in #quantum field theory are "perturbative", that means that one computes a series expansion in powers of some parameter, called g. The usual intuition is that the parameter is small, so that higher powers of the parameter are negligible. However, in most cases, these series are divergent so that their sum is infinite for every non-zero value of the parameter g. This sometimes leads to confusion. The solution is to realize that there exist perfectly fine functions where a power series expansion is divergent. Not every function has a convergent series expansion. In the context of quantum field theory, one can see this explicitly in the case of a zero-dimensional theory. In that case, every #FeynmanDiagram is assigned the value 1 (which is a huge simplification). One can then solve the entire perturbation series. It is divergent, but it can equivalently be represented as a convergent integral. The conclusion is that quantum field theory delivers a finite, smooth function, but its perturbation series diverges since that function is imaginary for certain parameter ranges (which has physical interpretation as decay of particles). The mathematics is fine, the problem is that physics education is very much focused on polynomials and Taylor series, which leads to the wrong expectation that every physically sensible function should have a convergent Taylor series expansion. I recently gave a talk about this at PI, slides are available on my homepage https://paulbalduf.com/research/vector-asymptotics/
#physics #math

At the @EPSHEP2025 #physics #conference, I gave a talk about behaviour of #feynmandiagram s at large loop order, and that at 18 loops we are still not observing the leading asymptotic growth rate. The slides and various data sets are available from my website as always. https://paulbalduf.com/research/vector-asymptotics/

At the @EPSHEP2025 #physics #conference, I gave a talk about behaviour of #feynmandiagram s at large loop order, and that at 18 loops we are still not observing the leading asymptotic growth rate. The slides and various data sets are available from my website as always. https://paulbalduf.com/research/vector-asymptotics/

a jumble of Feynman diagrams and coloured shapes with a Bauhaus feel.

#linocut #printmaking #physics #Bauhaus #quantumMechanics #FeynmanDiagram #penguinDiagram #particlePhysics #sciart #WorldQuantumDay

*I am somewhat dubious of this date because a) I was trained to use the metric convention day/month/year and b) physicists use the reduced constant h-bar (that’s h divided by 2 pi) and then, cause it’s easier they just change units such that h-bar = 1 but it’s as good an excuse as any

🧵3/3

a jumble of Feynman diagrams and coloured shapes with a Bauhaus feel.

#linocut #printmaking #physics #Bauhaus #quantumMechanics #FeynmanDiagram #penguinDiagram #particlePhysics #sciart #WorldQuantumDay

*I am somewhat dubious of this date because a) I was trained to use the metric convention day/month/year and b) physicists use the reduced constant h-bar (that’s h divided by 2 pi) and then, cause it’s easier they just change units such that h-bar = 1 but it’s as good an excuse as any

🧵3/3

April 14 has been designated World Quantum Day in honour of Planck’s Constant which can be rounded to h~ 4.14×10−15 eV·s (and some folks write April 14 as 4/14*). Planck’s constant comes up a lot in quantum mechanics; for instance a photon’s energy is h times its frequency). So I thought I would share Feynman Bauhaus.
🧵1/n

#linocut #printmaking #physics #Bauhaus #quantumMechanics #FeynmanDiagram #penguinDiagram #particlePhysics #sciart #WorldQuantumDay

April 14 has been designated World Quantum Day in honour of Planck’s Constant which can be rounded to h~ 4.14×10−15 eV·s (and some folks write April 14 as 4/14*). Planck’s constant comes up a lot in quantum mechanics; for instance a photon’s energy is h times its frequency). So I thought I would share Feynman Bauhaus.
🧵1/n

#linocut #printmaking #physics #Bauhaus #quantumMechanics #FeynmanDiagram #penguinDiagram #particlePhysics #sciart #WorldQuantumDay

Printed a new edition of my ‘Feynman Diagram Bauhaus’ print!

It’s was inspired by the way the exuberant lines and shapes of Bauhaus paintings, like those of Wassily Kandinsky, reminded me Feynman diagrams showing particle reactions on the quantum scale, virtual particles and anti-particles popping out of the vacuum on borrowed mass-energy and of quantum foam.

1/2

#linocut #printmaking #physics #quantumMechanics #FeynmanDiagram #MastoArt #particlePhysics #bauhaus #sciart

Printed a new edition of my ‘Feynman Diagram Bauhaus’ print!

It’s was inspired by the way the exuberant lines and shapes of Bauhaus paintings, like those of Wassily Kandinsky, reminded me Feynman diagrams showing particle reactions on the quantum scale, virtual particles and anti-particles popping out of the vacuum on borrowed mass-energy and of quantum foam.

1/2

#linocut #printmaking #physics #quantumMechanics #FeynmanDiagram #MastoArt #particlePhysics #bauhaus #sciart

#Inktober2021, Day 17: Prompt word: #collide

Physics time! Today: A #FeynmanDiagram of an #electron colliding with a #proton.
For (a LOT) more information: https://www.feynmanlectures.caltech.edu

Faber-Castell Pitt Artist Pen 0.5, Clairefontaine Flying Spirit sketch book
#Inktober