There is a new book about Lattice QCD, Lattice Quantum Chromodynamics: Practical Essentials by Francesco Knechtli, Michael Günther and Mike Peardon. At a 140 pages, this is a pretty slim volume, so it is obvious that it does not aim to displace time-honoured introductory textbooks like Montvay and Münster, or the newer books by Gattringer and Lang or DeGrand and DeTar. Instead, as suggested by the subtitle "Practical Essentials", and as said explicitly by the authors in their preface, this book aims to prepare beginning graduate students for their practical work in generating gauge configurations and measuring and analysing correlators.
In line with this aim, the authors spend relatively little time on the physical or field theoretic background; while some more advanced topics such as the Nielson-Ninomiya theorem and the Symanzik effective theory or touched upon, the treatment of foundational topics is generally quite brief, and some topics, such as lattice perturbation theory or non-perturbative renormalization, are altogether omitted. The focus of the book is on Monte Carlo simulations, for which both the basic ideas and practically relevant algorithms — heatbath and overrelaxation fro pure gauge fields, and hybrid Monte Carlo for dynamical fermions — are described in some detail, including the RHMC algorithm and advanced techniques such as determinant factorizations, higher-order symplectic integrators, and multiple-timescale integration. The techniques from linear algebra required to deal with fermions are also covered in some detail, from the basic ideas of Krylov space methods through concrete descriptions of the GMRES and CG algorithms, along with such important preconditioners as even-odd and domain decomposition, to the ideas of algebraic multigrid methods. Stochastic estimation of all-to-all propagators with dilution, the one-end trick and low-mode averaging and explained, as are techniques for building interpolating operators with specific quantum numbers, gauge link and quark field smearing, and the use of the variational method to extract hadronic mass spectra. Scale setting, the Wilson flow, and Lüscher's method for extracting scattering phase shifts are also discussed briefly, as are the basic statistical techniques for data analysis. Each chapter contains a list of references to the literature covering both original research articles and reviews and textbooks for further study.
Overall, I feel that the authors succeed very well at their stated aim of giving a quick introduction to the methods most relevant to current research in lattice QCD in order to let graduate students hit the ground running and get to perform research as quickly as possible. In fact, I am slightly worried that they may turn out to be too successful, since a graduate student having studied only this book could well start performing research, while having only a very limited understanding of the underlying field-theoretical ideas and problems (a problem that already exists in our field in any case). While this in no way detracts from the authors' achievement, and while I feel I can recommend this book to beginners, I nevertheless have to add that it should be complemented by a more field-theoretically oriented traditional textbook for completeness.
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Note that I have deliberately not linked to the Amazon page for this book. Please support your local bookstore — nowadays, you can usually order online on their websites, and many bookstores are more than happy to ship books by post.
In line with this aim, the authors spend relatively little time on the physical or field theoretic background; while some more advanced topics such as the Nielson-Ninomiya theorem and the Symanzik effective theory or touched upon, the treatment of foundational topics is generally quite brief, and some topics, such as lattice perturbation theory or non-perturbative renormalization, are altogether omitted. The focus of the book is on Monte Carlo simulations, for which both the basic ideas and practically relevant algorithms — heatbath and overrelaxation fro pure gauge fields, and hybrid Monte Carlo for dynamical fermions — are described in some detail, including the RHMC algorithm and advanced techniques such as determinant factorizations, higher-order symplectic integrators, and multiple-timescale integration. The techniques from linear algebra required to deal with fermions are also covered in some detail, from the basic ideas of Krylov space methods through concrete descriptions of the GMRES and CG algorithms, along with such important preconditioners as even-odd and domain decomposition, to the ideas of algebraic multigrid methods. Stochastic estimation of all-to-all propagators with dilution, the one-end trick and low-mode averaging and explained, as are techniques for building interpolating operators with specific quantum numbers, gauge link and quark field smearing, and the use of the variational method to extract hadronic mass spectra. Scale setting, the Wilson flow, and Lüscher's method for extracting scattering phase shifts are also discussed briefly, as are the basic statistical techniques for data analysis. Each chapter contains a list of references to the literature covering both original research articles and reviews and textbooks for further study.
Overall, I feel that the authors succeed very well at their stated aim of giving a quick introduction to the methods most relevant to current research in lattice QCD in order to let graduate students hit the ground running and get to perform research as quickly as possible. In fact, I am slightly worried that they may turn out to be too successful, since a graduate student having studied only this book could well start performing research, while having only a very limited understanding of the underlying field-theoretical ideas and problems (a problem that already exists in our field in any case). While this in no way detracts from the authors' achievement, and while I feel I can recommend this book to beginners, I nevertheless have to add that it should be complemented by a more field-theoretically oriented traditional textbook for completeness.
___
Note that I have deliberately not linked to the Amazon page for this book. Please support your local bookstore — nowadays, you can usually order online on their websites, and many bookstores are more than happy to ship books by post.
bath through the box’s walls. After exchanging heat for a long time, the gas reaches thermal equilibrium: Large-scale properties, such as the gas’s energy, quit changing much. Imagine measuring the gas’s energy. What probability does the measurement have of outputting 
? The Gibbs distribution provides the answer, 
. The 
denotes Boltzmann’s constant, a fundamental constant of nature. The 
denotes a partition function, which ensures that the probabilities sum to one.
. You’ll have poked the system’s temperature out of equilibrium. Heat will flow from the hotter bath to the colder bath. Particles carry the heat, energy of motion. Suppose that the particles have electric charges. An electric current will flow because of the temperature difference. Similarly, heat can flow because of an electric potential difference, or a pressure difference, and so on. You can cause a thermodynamic system’s elbow to itch, Onsager showed, by tickling the system’s ankle.
Calculating the tensor products of matrices is impractical because the sizes of the matrices increase exponentially with the number of qubits. An alternative that scales better with an increasing amount of qubits are graphical languages. A graphical language consists of a set of generators, some ways to compose them into a diagram, and a set of rules that tell you when two diagrams are equal. The dominant graphical language in quantum computation is the 
The circuit model makes tracking and simplifying a computation easier.
For instance take the circuit to the right: knowing that , and commute and , one can transform this circuit into a simpler one. 
These rules propose a systematic way for simplification of diagrams. For instance, using naturality of braiding in a symmetric category, we can simplify this diagram. 
The ZX-calculus is a PROP: a category of natural numbers with addition as a monoidal structure. There are only two types of morphism, red spiders and green spiders, which are decorated with an angle in and can have an arbitrary amount of inputs and outputs (in this picture, red node is a node with white and green node is the node with black ).
This rule cannot be derived from the categorical axioms; nonetheless, it is necessary for the completeness result. 
These generators are given by assigning an operator to each node and a operator to nodes connected to it. For instance, the generators of this graph are: 
. For instance, with regards to length 5 sign patterns 
of the 
possible sign patterns in 
occur with positive upper density. (Conjecturally, all of them do so, and this is known for all shorter sign patterns, but unfortunately 
, where 
is the number of prime factors of 
(counting multiplicity). One can also consider the variant 
, which is a completely multiplicative function taking values in the cube roots of unity 
. Here we are able to show that all 
sign patterns in 
occur with positive lower density as sign patterns 
of this function. The analogous result for 
(from 
case by itself (largely because the “parity” arguments used in the case of the Liouville function no longer control three-point correlations in the 
for a certain partition 
of a compact abelian group 
(think for instance of the unit circle 
, although the general case is a bit more complicated, in particular if 
, also we can allow the 
to be replaced by any 
with 
divisible by 
), with each of the 
having measure 
. An inequality of Kneser just barely fails to guarantee the existence of such patterns, but by using an inverse theorem for Kneser’s inequality in 
of a natural number 
(this improves on a 
replaced by 
. A number of other results of this type are also obtained in this paper.
which have some multiplicative structure, in the sense that (roughly speaking) there is a set 
such that for all small primes 
, the statements 
and 
are roughly equivalent to each other. For instance, if 
, one would take 
; if instead 
, then one can take 
. When one has such a situation, then very roughly speaking, the entropy decrement argument then allows one to estimate a one-parameter correlation such as 
denote the vector space of polynomials 
of one variable 
with real coefficients of degree at most 
, and the sequence of these spaces form a 
: every polynomial 
in 
. More generally, if one has any sequence 
of polynomials, with each 
of degree exactly 
forms a basis for 
and 
of polynomials, with each 
of degree 
, then 
, thus we have an identity of the form 
. These coefficients describe how to convert a polynomial expressed in the 
involving two natural numbers 
can be interpreted as such change of basis coefficients. The most familiar example are the binomial coefficients 
, which measures the conversion from the shifted monomial basis 
to the monomial basis 
, thanks to (a special case of) the binomial formula: 
, the conversion from 
to 
, thanks to the general case of the binomial formula.
: 
: 
as a basis instead of the falling factorials, one of course can rewrite these conversions as 
of the first kind: 
of degree 
. 
. 
are the number of permutations of 
] and in particular about the frustration of not finding any helpful big surprises during this great decade of fundamental science — but for now, let’s just enjoy this incredible image for what it is, and congratulate those who proposed this effort and those who carried it out.
where 
is the free strict symmetric monoidal category on a set 
Elements of 
represent different kinds of ‘agents’. Unlike in a Petri net, we do not usually consider processes where these agents turn into other agents. Instead, we wish to study everything that can be done with a fixed collection of agents. Any object 
is of the form 
for some 
thus, it describes a collection of agents of various kinds. The functor 
maps this object to a category 
that describes everything that can be done with this collection of agents.
Composing these morphisms corresponds to ‘overlaying’ graphs. Network models of this sort let us design networks where the nodes are agents and the edges are communication channels or shared commitments. In our first paper the operation of overlaying graphs was always commutative:
can carry out. Composition of morphisms corresponds to carrying out first one process and then another.
determines a symmetric monoidal category 
whose morphisms are processes that can be carried out using this Petri net. More precisely, the objects in 
and the morphisms are sequences of ways to change these markings using transitions, e.g.:
is a catalyst:
nor 
is a catalyst. The transition 
requires one token of type 
requires no tokens of type 
by ‘gluing together’ all the categories 
but it is better suited to applications where agents each have their own ‘individuality’, because 
when 
since no transitions can change the amount of catalysts present. The category 
where 
an element of free commutative monoid on 
specifies the amount of each catalyst present.
gives an object in 
and similarly for morphisms. However, in Theorem 14 we show that each subcategory 
represents the catalysts, and 
and 
are two processes which we can carry out using these catalysts. We can do either one first, but we get different morphisms as a result.
for varieties 
(or similar objects) defined over arbitrary commutative rings 
, and with coefficients in another arbitrary commutative ring 
or 
, but can allow for arbitrary coefficients 
-adic cohomology
is localised around a prime 
. The following picture Scholze drew in his first lecture captures these sorts of relationships nicely:
in the above diagram, in which the domain ring 
of these rings is either zero, or “small”. For instance, the 
is technically of characteristic 
, but 
is a “small” ideal of 
), so one can think of 
for which prismatic cohomology is effective:
one needs a “prism”: a ring homomorphism from 
on 
-deformation
that for instance applied to monomials 
by the usual formula
that for instance applies to monomials ![\displaystyle d_q (t^n) = [n]_q t^{n-1} d_q t](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+d_q+%28t%5En%29+%3D+%5Bn%5D_q+t%5E%7Bn-1%7D+d_q+t&bg=ffffff&fg=000000&s=0 "\displaystyle d_q (t^n) = [n]_q t^{n-1} d_q t")
![\displaystyle [n]_q = \frac{q^n-1}{q-1} = 1 + q + \dots + q^{n-1}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Bn%5D_q+%3D+%5Cfrac%7Bq%5En-1%7D%7Bq-1%7D+%3D+1+%2B+q+%2B+%5Cdots+%2B+q%5E%7Bn-1%7D&bg=ffffff&fg=000000&s=0 "\displaystyle [n]_q = \frac{q^n-1}{q-1} = 1 + q + \dots + q^{n-1}")
). (The 
