Site Log

• Date:  2024-07-11

  Updated to version 2024-07-11:
  50710 (rewrote the section about dimension-dependence to clarify the reasoning, adjusted the wording in the section about the shell theorem)

• Date:  2024-06-23

  Updated to version 2024-06-23:
  00418 (changed the wording in a few places to avoid using the overloaded term "gauge symmetry")
  03723 (fixed a spelling error)
  21916 (removed an unnecessary assertion about the meaning of gauge symmetry)
  37301 (removed the statements about gauge symmetry -- which didn't add any technical insight anyway -- to avoid inconsistency with other usages of the words)
  51033 (clarified the order in which the continuous-time limit and Wick rotation should be applied, added a reference that explains why the order matters, and added a footnote to clarify a step in the derivation of the hamiltonian for principal chiral models)
  51376 (added a footnote about normal operators and observables, replaced "gauge invariant" with a more specific term as a reminder that we only require invariance under gauge transformations that act trivially on the boundary, modified the hamiltonian to include plaquettes that touch the boundary when the lattice is truncated, and updated the wording in several sections to support that modification)
  63548 (changed a footnote about unitarity in other models and added a reference to an article that was written later)
  78463 (clarified a footnote about gauge invariance and observables)

• Date:  2024-06-01

  Updated to version 2024-06-01:
  51376 (replaced a confused paragraph about two different continuum limits in two-dimensional space and added a reference that compares those different continuum limits)

• Date:  2024-05-26

  New article, version 2024-05-26:
  11617

• Date:  2024-05-24

  Updated to version 2024-05-24:
  03431 (added a sentence to the abstract)

• Date:  2024-05-22

  Updated to version 2024-05-22:
  20554 (changed the title and abstract to be more descriptive)

• Date:  2024-05-21

  Updated to version 2024-05-21:
  03431, 24902, 29682, 31738, 32191, 36176, 51186, 54711, 66313, 70833, 73054, 77228, 81674, 92035 (changed the titles and some of the abstracts to be more descriptive)

• Date:  2024-05-19

  Updated to version 2024-05-19:
  28539 (adjusted two section titles, moved one statement to a footnote, added a derivation of the homology groups of an n-dimensional torus)
  51376 (added a reference)
  61813 (added a reference, moved some content from the section about homotopy sets to a new article)
  70621 (added another citation to an already-cited reference)
  92035 (added a reference)

• Date:  2024-05-19

  New articles, all version 2024-05-19:
  33600, 35490, 69958

• Date:  2024-05-02

  Updated to version 2024-05-02:
  61813 (cited two more references, added a missing word "connected")
  70621 (fixed a sentence that conflated local trivialization functions and transition functions in the section about constructing a fiber bundle from trivial patches)

• Date:  2024-04-25

  Updated to version 2024-04-25:
  61813 (added a phrase with "basepoint" to a sentence in the first paragraph about higher homotopy groups, added a section about homotopy sets with and without basepoints, removed one sphere example after finding a discrepancy in the literature, added two more sphere examples, moved a footnote that was tied to the wrong sentence)

• Date:  2024-04-20

  Updated to version 2024-04-20:
  28539 (added references to a book by Whitehead, added a result about the nth homology group of an n-dimensional manifold, added results related to Hurewicz isomorphism, added definition of homology equivalence)
  61813 (separated the cartesian-product example into its own section and added more results, added more sphere examples)
  70621 (reworded the comments about generalizing to infinite-dimensional spaces, added the condition that transition functions should be restricted to a Lie group of diffeomorphisms of the fiber, added a footnote about the condition for the bundle to be trivial, fixed a broken sentence in the section about two meanings of "structure group", added a reference for the definition of local section)
  92035 (changed equals to isomorphic in one equation, added a note about Bott periodicity)

• Date:  2024-04-16

  Updated to version 2024-04-16:
  09894 (replaced "a identity element" with "an identity element")
  28539 (expanded the section about notation, separated the section about products and sums into two sections and expanded the tensor product section, reworded the comparison between bordism homology and singular homology, separated the section about torsion into a section about abelian groups and a section about compact manifolds and slightly expanded both of them, added a section reviewing the definitions of ring and principal ideal domain and field, added sections about cohomology groups and cohomology rings, added sections about (co)homology with coefficients in a field, added a section about cartesian products of spheres)
  61813 (added a cross-reference for the direct product of two groups, updated publication info for one reference)
  92035 (added another definition of semisimple for compact connected Lie groups, added a footnote defining semisimple for not-necessarily-compact Lie groups, added a section with examples for a general structure theorem, added a section about maximal tori and rank, added a sentence relating the rank to the notation, added sections about homology groups and torsion, added a section about cohomology rings and their relationship to homology groups, changed "group" to "compact topological group" in the footnote about Tannaka-Krein duality and removed an unhelpful reference)

• Date:  2024-04-08

  New article, version 2024-04-08:
  92035

• Date:  2024-04-01

  Updated to version 2024-04-01:
  29682 (added a footnote about terminology)
  70621 (fixed the wording of one phrase in the definition of principal G-bundle to clarify that the G-action on the fiber must be free and transitive)

• Date:  2024-03-24

  Updated to version 2024-03-24:
  44113 (corrected spelling Steifel -> Stiefel, added a reference about Stiefel-Whitney numbers and boundaries, added a reference about another definition of linking number)
  51376 (corrected a typo)
  61813 (moved one section to a better place in the sequence, added citations to new references, changed equal -> isomorphic in a few statements about homotopy groups)
  93875 (added sections reviewing the concept of a CW complex, added a section about other generalizations)

• Date:  2024-03-09

  Updated to version 2024-03-09:
  51033 (clarified the relationship between two equations that use different notation)

• Date:  2024-03-08

  Updated to version 2024-03-08:
  51033 (made a footnote about Wick rotation more careful, fixed a notation inconsistency)
  63548 (added a footnote about making the transfer matrix more symmetric)
  76708 (adjusted the wording of two sentences to improve readability)

• Date:  2024-03-07

  Updated to version 2024-03-07:
  03431 (added a reference)
  21794 (added a reference to a review of antilinear operators)
  51376 (fixed statements involving Stokes's theorem so that they remain valid when space is not contractible)
  70621 (added the word "smooth" in two sentences in the chapter about sections compared to fiber-valued functions)

• Date:  2024-03-04

  Updated to version 2024-03-04:
  08264 (updated publication info for one reference)
  09894 (Stokes' theorem -> Stokes's theorem)
  19253 (non-abelian -> nonabelian)
  21916 (updated publication info for one reference)
  26542 (fixed the units in an allusion to compact lattice QED, added cross-references to a new article)
  29682 (updated publication info for one reference)
  32191 (non-abelian -> abelian)
  37301 (updated publication info for one reference)
  70621 (updated publication info for one reference, added a footnote about distinguishing between data and conditions)
  76708 (Stokes' -> Stokes's, and fixed the grammar in one sentence)
  80838 (fixed a grammar error)
  81040 (updated publication info for one reference)

• Date:  2024-03-04

  New article, version 2024-03-04:
  51376

• Date:  2024-03-03

  Updated to version 2024-03-03:
  26542 (consolidated two redundant footnotes, polished one sentence, added a notation example, fixed a broken explanation in one footnote, moved one distracting sentence to a footnote, added a comparison to scalar fields in a footnote)
  31738 (reformatted an equation, added references for the iconic form, added a footnote about conventions)
  54711 (fixed an index typo, added references for the traditional form)
  78463 (added an equation to make the convention for the spacetime metric more prominent)

• Date:  2024-03-02

  New article, version 2024-03-02:
  26542

• Date:  2024-02-27

  Updated to version 2024-02-27:
  44113 (removed an unused reference)
  61813 (removed unused references, added a missing reference)

• Date:  2024-02-25

  Updated to version 2024-02-25:
  29682 (added another example of isomorphism)
  70621 (tweaked the font used to denote real projective space)
  76708 (added some references about expressing connections in terms of holonomies)
  93875 (replaced a sentence in the abstract with a more informative one, added references about topological manifolds, added a section relating metric spaces to manifolds, refined the definition of smooth structure to be maximal smooth atlas, added more references in the section about diffeomorphisms, clarified the dimensions in the section about diffeomorphisms, removed a ill-conceived reason in the section about diffeomorphisms, added a clarification in the section about diffeomorphisms, added new sections about relationships between different special kinds of topological spaces)
  03431, 08264, 10142, 33547, 36176, 48968, 70621, 77228, 79649, 81040, 81674, 98038 (fixed incorrect grammar: "whether or not" -> "whether" when the outcome is in question)

• Date:  2024-02-25

  New articles, all version 2024-02-25:
  28539, 44113, 61813

• Date:  2023-12-23

  Updated to version 2023-12-23:
  99922 (replaced B<1 with B>1 in the sentence about conical singularity, added a footnote explaining how to deduce the effect of the conical singularity)

• Date:  2023-12-16

  New article, version 2023-12-16:
  03723

• Date:  2023-12-16

  Updated to version 2023-12-16:
  11475 (clarified the wording of a footnote)
  96560 (clarified the definition of conformally flat)

• Date:  2023-12-11

  Updated to version 2023-12-11:
  03519 (added a paragraph about the symmetry of the Ricci tensor, added another terminology variation)
  96560 (updated a paragraph about notation in the introduction, added a sentence acknowledging that dS and AdS spacetimes are examples)

• Date:  2023-12-10

  Updated to version 2023-12-10:
  96560 (generalized the sphere example to a family of maximally symmetric spaces, added the result for the full Ricci tensor instead of only the diagonal components, fixed an incorrect coefficient in the result for the Ricci tensor, fixed an index-notation collision, updated the abstract)

• Date:  2023-12-09

  Updated to version 2023-12-09:
  03519 (split the section about the curvature tensor into smaller sections, revised it to use a fixed sign convention for the Ricci tensor that seems to be standard in the physics literature, moved the curvature-of-sphere sign-check to a separate article devoted to sign conventions)
  24902 (added a reference, and added a cross-reference to a new article in this series)

• Date:  2023-12-09

  New articles, all version 2023-12-09:
  80838, 96560, 99922

• Date:  2023-11-12

  Updated to version 2023-11-12:
  03431 (moved a reference from one footnote to a more appropriate footnote, and added a reference)

• Date:  2023-11-12

  Updated to version 2023-11-12:
  03838, 09193, 21916, 28477, 37301, 41522, 52890 (made project euclid url shorter)
  00980, 09193, 15939, 20554, 21916, 22792, 22871, 28477, 30983, 37301, 41522, 44563, 52890, 81040, 85870, 89695 (replaced Hamiltonian with hamiltonian)
  12342, 63548 (replaced Lagrangian with lagrangian)
  09894, 10254, 38111, 49705 (replaced Lorentzian with lorentzian)
  09894, 10254, 12883, 21808, 37301, 38111, 93875 (replaced Euclidean with euclidean)

• Date:  2023-10-26

  Updated to version 2023-10-26:
  21916 (added a few new references)
  27882 (added a reference)
  51033 (removed a dubious-and-unnecessary statement)
  70621 (deleted a stray subscript, deleted a stray parenthesis, added some vocabulary)
  78463 (added a footnote about units)

• Date:  2023-10-05

  Updated to version 2023-10-05:
  03838 (expanded the description of other circle bundles, added a footnote, and added a reference)
  70621 (corrected a statement in a caption about the significance of the dashed lines, fixed a notation discrepancy in a figure, added a footnote about line bundles, added a reference)

• Date:  2023-10-01

  Updated to version 2023-10-01:
  09894 (added a reference)
  86175 (fixed a copy-paste error, clarified notation in one section, changed one section title)
  93875 (separated two facts that were incorrectly conflated in the section about diffeomorphisms)

• Date:  2023-10-01

  New articles, all version 2023-10-01:
  70621, 76708, 03838

• Date:  2023-06-05

  Updated to version 2023-06-05:
  03431 (fixed a typo, fixed the format of five footnotes, removed a hyphen, improved the wording of two sentences without changing the meaning)

• Date:  2023-06-04

  New article, version 2023-06-04:
  27882

• Date:  2023-06-02

  New article, version 2023-06-02:
  87696

• Date:  2023-06-02

  Updated to version 2023-06-02:
  08264 (made the abstract more precise)
  21794 (replaced some references with cross-references to a new article)
  86175 (added a footnote with another warning about language, made the abstract more precise)

• Date:  2023-05-28

  New article, version 2023-05-28:
  21794

• Date:  2023-05-28

  Updated to version 2023-05-28:
  32191 (added a missing item to the bibliography)
  86175 (deleted an extra word, clarified a footnote, added a sentence about hermitian/antihermitian)

• Date:  2023-05-14

  Updated to version 2023-05-14:
  08264 (added two footnotes)
  09894 (fixed a typo)
  86175 (added a cross-reference to the abstract, reworded one phrase for clarity, fixed an incorrect word (odd → even) in one section and separated the section into two to improve readability)
  93169 (added a missing word)

• Date:  2023-05-08

  New article, version 2023-05-08:
  86175

• Date:  2023-05-08

  Updated to version 2023-05-08:
  03910 (added a phrase to clarify one definition, added a footnote about another isomorphism, added a warning about an error (a mis-typed equation) in one reference)
  08264 (deleted a paragraph that didn't add any insight, moved a cross-reference)

• Date:  2023-04-30

  New article, version 2023-04-30:
  08264

• Date:  2023-04-30

  Updated to version 2023-04-30:
  03910 (added note about a sign convention, with a reference)
  39430 (added a reference)

• Date:  2023-04-21

  New article, version 2023-04-21:
  03910

• Date:  2023-04-21

  Updated to version 2023-04-21:
  07246 (removed a detail that isn't explicitly stated in the cited paper)

• Date:  2023-04-17

  New articles, both version 2023-04-17:
  07246, 51033

• Date:  2023-04-17

  Updated to version 2023-04-17:
  21916 (fixed a typo in the bibliography)
  81040 (added the page-range to an item in the bibliography)

• Date:  2023-02-12

  Updated to version 2023-02-12:
  18505 (removed an incorrect statement from the new footnote)

• Date:  2023-02-10

  Updated to version 2023-02-11:
  79649 (replaced a link in a footnote with a proper citation)
  79649v.py, 79649s.py (re-posted the python scripts to restore part of a line that was made invisible by the html interpreter)
  79649f.py (re-posted the python script to remove a temporary change of axes)

• Date:  2023-02-10

  Updated to version 2023-02-10:
  03431 (added references about one attempt to derive Born's rule and a reference about MWI)
  09894 (added a reference)
  18505 (added a footnote about an alternative definition of eigenvalue)
  21916 (added a cross-reference to a new article)
  22050 (specified a range of n in the section about evaluating the rotation-symmetric term)
  37301 (added a reference)
  81040 (deleted an extra word in a footnote, added a reference)

• Date:  2023-02-10

  New articles, all version 2023-02-10:
  23277, 63548, 10142, 22212, 79649

• Date:  2022-10-23

  Updated to version 2022-10-23:
  03431, 11475, 21916, 22871, 23206, 24902, 32191, 33547, 36176, 48968, 70833, 74088, 77228, 78463, 81040, 90771 (updated bibliography to include publication data instead of only arxiv link)

• Date:  2022-10-22

  New articles, both version 2022-10-22:
  22792, 85870

• Date:  2022-10-15

  New articles, both version 2022-10-15:
  09193, 37301

• Date:  2022-09-27

  New article, version 2022-09-27:
  10254

• Date:  2022-09-27

  Updated to version 2022-09-27:
  38111 (corrected the indices in one equation)

• Date:  2022-09-26

  Updated to version 2022-09-26:
  38111 (clarified the scope of a few other statements in the text)

• Date:  2022-09-25

  Updated to version 2022-09-25:
  38111 (added a missing equation-reference, removed part of one footnote, corrected the scope of a statement in the abstract, added a new section about the concept of an induced metric, clarified the purpose of the group-theory section)

• Date:  2022-09-24

  New article, version 2022-09-24:
  38111

• Date:  2022-09-24

  Updated to version 2022-09-24:
  44563 (fixed an equation-reference)

• Date:  2022-08-23

  Updated to version 2022-08-23:
  00980 (added a cross-reference)
  22050 (deleted an extra phrase from the section "Evaluating the remainder")
  30983 (added clarification about the context of the "position operator")
  58590 (moved a paragraph break to improve readability)
  98038 (changed the font used for energy to avoid conflict with the letter E for electric field, added a conclusion-paragraph in section 13)

• Date:  2022-08-21

  New articles, all version 2022-08-21:
  00980, 22050, 30983, 44563, 52890, 58590, 98038

• Date:  2022-08-21

  Updated to version 2022-08-21:
  28477 (added a reference, fixed an invisible character in the bibliography)
  71852 (expanded the section about the Fourier transforms and moved part of it into the section about the low-momentum approximation)

• Date:  2022-06-27

  Updated to version 2022-06-28:
  81040 (fixed the proof that local observables can't mix the symmetry-breaking ground states, added a footnote about the cluster property)

• Date:  2022-06-26

  New article, version 2022-06-26:
  81040

• Date:  2022-06-18

  Updated to version 2022-06-18:
  36176 (removed a duplicate word, added a footnote to clarify some standard-but-overloaded notation, added a note to define some other standard notation)

• Date:  2022-06-18

  New article, version 2022-06-18:
  12883

• Date:  2022-06-13

  Updated to version 2022-06-13:
  36176 (removed unused abbreviation from abstract)
  71852 (deleted an extra word in the abstract)

• Date:  2022-06-11

  Re-posted all articles so that references to articles in this series link to html abstract pages instead of to pdfs. Didn't change any version numbers/dates.

• Date:  2022-06-07

  Updated to version 2022-06-07:
  28477 (generalized an unnecessarily-specific equation reference)

• Date:  2022-06-04

  Updated to version 2022-06-05:
  28477 (simplified the equation that characterizes local observables)
  Updated to version 2022-06-04:
  00669 (deleted an extra word, moved one definition to an earlier section)
  41522 (added new section, chose a different letter for one operator)
  89695 (added new section, chose a different letter for one operator, removed a hyphen)

• Date:  2022-06-04

  New articles, all version 2022-06-04:
  15939, 28477, 71852

• Date:  2022-05-25

  Updated to version 2022-05-25:
  89695 (rewrote the section "Detection observables for composite particles" to be a little more explicit)

• Date:  2022-05-24

  Updated to version 2022-05-24:
  41522 (trimmed abstract, added a synonym in a footnote, fixed two typos, added an equation cross-reference, adjusted wording in two sentences)
  89695 (replaced an incorrect characterization of local observables in the "Perspective and approach" section, improved wording in one sentence, fixed a punctuation typo, added a missing condition in a footnote, extended the section "Detection observables for composite particles" to address detection probabilities)

• Date:  2022-05-23

  Updated to version 2022-05-23:
  20554 (added two sentences about the Schrödinger picture, removed one unimportant footnote to make room for two new sentences about other observables)
  36176 (added footnotes about etymology and pronunciation)

• Date:  2022-05-23

  New articles, all version 2022-05-23:
  41522, 89695

• Date:  2022-05-14

  Updated to version 2022-05-15:
  36176 (improved some wording, fixed grammar errors, moved an inline reference to the bibliography, removed part of one footnote)

• Date:  2022-05-14

  New article, version 2022-05-14:
  36176

• Date:  2022-05-09

  Updated to version 2022-05-09:
  20554 (minor cleanup, added some clarifications, replaced one example)

• Date:  2022-05-08

  Updated to version 2022-05-08:
  22871 (removed an incorrect reference, corrected another reference, corrected the statement of Stone's theorem)

• Date:  2022-05-08

  New article, version 2022-05-08:
  20554

• Date:  2022-04-30

  Updated to version 2022-04-30:
  03431 (deleted a redundant sentence, fixed an out-of-sync phrase, reworded a few sentences, clarified one footnote)
  21916 (removed a broken cross-reference, clarified one sentence, rearranged one paragraph)
  22871 (fixed a spelling error)
  70833 (deleted one word, and fixed a cross-reference)
  74088 (updated/added references, added a footnote, genralized one theorem, changed one word)
  77228 (added two new footnotes)
  90771 (added a new section about bases, rewrote the section about separability, updated/added some references)
  Also:
  Re-posted all pdf files to change the color of hyperlinks from black to blue, to clarify the fact that the hyperlinks actually do something. Didn't change any version numbers/dates.

• Date:  2022-04-30

  New articles, all version 2022-04-30:
  03431, 21916, 22871, 70833

• Date:  2022-03-01

  Updated to version 2022-03-02:
  09894 (added a paragraph about different meanings of the word "field," with a new reference)
  74088 (removed an out-of-place phrase about components, reworded a confusing sentence about eigenvectors, and rearranged a sentence about notation)
  90771 (fixed inconsistent usage of the word "norm")

• Date:  2022-03-01

  New articles, all version 2022-03-01:
  74088, 77228, 90771

• Date:  2022-02-20

  New article, version 2022-02-20:
  39430

• Date:  2022-02-20

  Updated to version 2022-02-20:
  00418 (changed two equation-references and removed an extra syllable in one word)
  00669 (clarified the wording in a footnote)
  18505 (in the null-rotation example: clarified one sentence and added an equation)
  78463 (added a paragraph about the importance of relative signs, added a footnote about notation, and added boxes around some equations)

• Date:  2022-02-19

  New articles, all version 2022-02-18:
  00418, 00669, 31738, 54711, 78463, 98002

• Date:  2022-02-19

  Updated to version 2022-02-18:
  09894 (added a reference, fixed a spelling error)
  18505 (added a null-rotation example, reformatted to avoid a page break)
  29682 (added more references)
  32191 (added a cross-reference)
  37501 (added a cross-reference)
  41182 (replaced a cross-reference with a better one)
  71500 (added a cross-reference, fixed a spelling error)
  77597 (added a section about adding speeds)

• Date:  2022-02-07

  New article, version 2022-02-07:
  29682

• Date:  2022-02-06

  Updated to version 2022-02-06:
  03519 (fixed incorrect statement about metric-dependence in the context of sign conventions)
  18505 (fixed inconsistent notation in derivation of derivative of inverse matrix)
  32191 (added footnote to explain a detail in the charged scalar field example)
  81674 (reformatted title)

• Date:  2022-02-05

  New articles, all version 2022-02-05:
  03519, 11475, 18505, 19253, 32191, 37501, 49705, 71500

• Date:  2022-02-05

  Updated to version 2022-02-05:
  09894 (added a footnote about notation)
  12342 (fixed pdf metadata, reformatted title)
  33547 (fixed pdf metadata, added new cross-references, fixed a sign error in the equation for the coefficients of the general geodesic equation)
  33629 (fixed pdf metadata, reformatted title, improved phonetic spelling in a footnote)
  41182 (reformatted title)
  46044 (fixed pdf metadata, reformatted title)
  93875 (fixed pdf metadata, reformatted title)
  Fixed pdf metadata: 12707, 23206, 24902, 37431, 50710, 51186, 66313, 73054, 81674, 93169

• Date:  2022-01-18

  New articles, all version 2022-01-16:
  09894, 12342, 12707, 21808, 23206, 24902, 33547, 33629, 37431, 41182, 46044, 48968, 50710, 51186, 66313, 73054, 77597, 81674, 93169, 93875