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a0e48c5485c2513a826a9f61d5f1430d15e74a6c | subsection | 47 | 110 | Linear sets and functions | (x\in U).Remark
As another concrete advantage of an involutive negation, recall that classically we can express “there is at most one x with P(x)” either as “for all x,y, if P(x) and P(y), then x=y” or “there do not exist x,y with x\ne y such that P(x) and P(y)”.
Intuitionistically these are no longer equivalent (unle... | {
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} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
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f717294db729ea9d16a7e9b9a8d09f1560d48efb | subsection | 48 | 110 | Linear sets and functions | ((x\circeq y) \boxtimes (y\mathrel {{\sqsubset }{\mathord {-}}}U))}\:\right.\\
\check{}\hspace{0.5pt}U &\mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}\protect \left.\:{x:A {\textstyle \sqcap }y^A. ((x\circeq y) \multimap (y\mathrel {{\sqsubset }{\mathord {-}}}U))}\:\right..The poset of subsets of A th... | {
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3efadf1330f469a6ed4f1f96f1ba6ca9a43c93db | subsection | 49 | 110 | Linear sets and functions | But if A has affirmative equality, does preserve subsets, hence =\hat{}\hspace{0.5pt}:(x\circeq y) \boxtimes (x\mathrel {{\sqsubset }{\mathord {-}}}U) \;\equiv \;
(x\circeq y) \boxtimes (x\mathrel {{\sqsubset }{\mathord {-}}}U) \;\equiv \;
((x\circeq y) \boxtimes (x\mathrel {{\sqsubset }{\mathord {-}}}U)) \;\vdash \;
(... | {
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229fd077eb5aefb4aeaabb3b412ef887aca85b59 | subsection | 50 | 110 | Linear sets and functions | This gives a semicartesian monoidal structure; the nontrivial part is associativity:\sharp (A \boxtimes \sharp (B\boxtimes C)) = \sharp (\sharp A \boxtimes \sharp (B\boxtimes C)) = \sharp (A\boxtimes (B\boxtimes C)) = \cdotsClosure under duality implies \ast -autonomy, with cotensor A \mathbin {\raisebox {-1pt}{\rotate... | {
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8576383ce1a98bcca3ccb029823b1f75c736695f | subsection | 51 | 110 | Linear sets and functions | ((x\circeq y) \multimap ((y\mathrel {{\sqsubset }{\mathord {-}}}U) \mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}}(y\mathrel {{\sqsubset }{\mathord {-}}}V)))}\:\right.\\
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f7065228503c2a1ba7d5cd36b5b7e9f3dc9b9b56 | subsection | 52 | 110 | Linear sets and functions | (x=y \wedge y\in U) \equiv (x\in U).And since {((y\in U))}^- \equiv \lnot (y\in U), the definition of \boxtimes gives us(x\in {\hat{}\hspace{0.5pt}U})\equiv \forall y. (((y\in U) \rightarrow (x\ne y)) \wedge ((x=y) \rightarrow \lnot (y\in U))),in which the second conjunct is superfluous.Remark
A subset that is an affi... | {
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dd7700618dd13611b228b00935d3552ee4741041 | subsection | 53 | 110 | Linear sets and functions | A refutative strong subset of such an -set is a strongly extensional -subset.Thus, intuitionistic constructive mathematics in the “sophisticated” sense, with tight apartness relations, embeds into linear constructive mathematics as the “refutative strong subuniverse”.Finally, we note that “anafunctions” (see thm:anafun... | {
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f6a5288d6ba1ae06bea59436b010364797740f50 | subsection | 54 | 110 | Linear sets and functions | \end{array}Theorem 6.8 In the standard interpretation, an -anafunction from A to B corresponds to an -anafunction that is “strongly extensional” in the sense thatF(x_1,y_1) \wedge F(x_2,y_2) \wedge (y_1\ne y_2) \vdash _{x_1,x_2\in A, y_1,y_2\in B} (x_1\ne x_2).An -anafunction consists of two -relations F,F on A\times B... | {
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b532ebb6732af19be8884a84c6325b25247d958d | subsection | 55 | 110 | Algebra | Roughly speaking, there are two approaches to intuitionistic constructive algebra.
The first uses apartness only minimally; inequality usually means denial \lnot (x=y) and is avoided as much as possible.
For instance, apartness relations are absent from , and are only rarely used in .
The second approach equips all set... | {
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a12ad132eee5e72b2b96d6576a5319e104cfe971 | subsection | 56 | 110 | Algebra | There are also natural examples in between; see eg:lpo.Definition
A group is an (-)set G together with an element e\mathchoice{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\mathrel {\raisebox {-1pt}{\script... | {
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8a064dff9bb46d0e2afb732b33e20f91cce9baa0 | subsection | 57 | 110 | Algebra | \end{array}The extra condition for G to be strong is(x u \ne y v) &\vdash _{x,y,u,v\in G} &\;& (x \ne y) \vee (u\ne v)which is equivalent to \ne being an apartness.
In particular:An -group with affirmative equality is precisely an -group.
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e5219c3c1b8d881aab52e3714effc204533f8d4e | subsection | 58 | 110 | Algebra | In fact, it can be proven purely in linear logic that an -group is strong if and only if it has strong equality.Definition A subgroup of a group G is a subset H\sqsubseteq G such that&\vdash &\;& (e\mathrel {{\sqsubset }{\mathord {-}}}H)\\
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2632ac6b74212908a494099468370638fe61f924 | subsection | 59 | 110 | Algebra | An affirmative -subgroup of an affirmative -group is precisely an -subgroup of an -group, together with its logical complement H \mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}{x\in G | \lnot (x\in H)}.
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faa391758936fe36cc84abc6973ad9f9555dc268 | subsection | 60 | 110 | Algebra | For the first, we have(x\circeq _H y) \boxtimes (w\circeq _H z)
&\equiv (x y^{-1} \mathrel {{\sqsubset }{\mathord {-}}}H)\boxtimes (w z^{-1} \mathrel {{\sqsubset }{\mathord {-}}}H)\\
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6c6026b26b6adef907de58775df10aaec2128104 | subsection | 61 | 110 | Algebra | ((x\mathrel {{\sqsubset }{\mathord {-}}}H) \multimap (f(x) \circeq e)) means not only (x\in H) \rightarrow (f(x)=e) but also (f(x)\ne e) \rightarrow (x\in H).Definition The kernel of a homomorphism f:G\rightarrow K is \protect \left.\:{x\in G f(x)\circeq e}\:\right..Theorem 7.6 The kernel H_f of f:G\rightarrow K is a n... | {
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85dfa7416bc04f62dbc71963a7b935c5813227f7 | subsection | 62 | 110 | Algebra | A general -ring is an -ring with an inequality such that (x\ne y) \leftrightarrow (x-y \ne 0) and (xy \ne 0) \rightarrow (y\ne 0).An ideal is an additive subgroup J with (x\mathrel {{\sqsubset }{\mathord {-}}}J) \vdash (xy\mathrel {{\sqsubset }{\mathord {-}}}J).
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8506ffcd7b5c4e0320d78444bc9422cd33d776c8 | subsection | 63 | 110 | Algebra | A \mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}}-prime strong refutative -ideal in a strong refutative -ring is an anti-ideal J in an -ring with apartness that is proper (1\in J) and such that (x\in J) \wedge (y\in J) \vdash (x y \in J), i.e. a prime anti-ideal as in .
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2f23002173bc23cb9ec1ccab97f27934f6fbd437 | subsection | 64 | 110 | Algebra | A general -ring is a \mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}}-field just when its corresponding -ring with inequality satisfies 0\ne 1 and (x\ne 0) \rightarrow \mathsf {inv}(x).
This is precisely a field as in with \ne irreflexive (in the zero ring is a “field” with 0\ne 0).
A \mathbin {... | {
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384cd3fbafd7c644a5bd9cbb0b6cd22311e20062 | subsection | 65 | 110 | Algebra | R is \mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}}-local if \mathsf {inv}(x+y) \vdash \mathsf {inv}(x) \mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}}\mathsf {inv}(y).In the standard interpretation:An affirmative \sqcup -local -ring is an -ring with \lnot (0=1) and \mathsf ... | {
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"raw": "P.T Johnstone. Rings, fields, and spectra. Journal of Algebra, 49(1):238 – 260, 1977.",
"source_ref_id": "de078927ed80a31ca71b04acd6... | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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4bf5f105e2072dc416bb64205f00d2cb5a75957f | subsection | 66 | 110 | Algebra | Similarly, the \sqcup -local condition contraposes to an instance of strong functionality.Finally, the \sqcup /\mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}} worlds get mixed a bit when we talk about residue fields:Theorem 7.9 If R is an \mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \bo... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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3dddb753d2f2338e42241e003dcdd6f18e1b5d4c | subsection | 67 | 110 | Order | When equality is defined, we can either introduce order and topology as structures on a pre-set which induce an equality, or as structures on a set that might determine the equality by a “separation” axiom.
In general we prefer the former.Definition A preorder on an (-)pre-set A is a predicate \sqsubseteq on A\times A ... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... |
05a437767e1b3792c67f02fbd912e3a9cddcd2c3 | subsection | 68 | 110 | Order | \end{array}If A has a preorder, then (x\circeq y) \mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}(x\sqsubseteq y) \sqcap (y\sqsubseteq x) makes A into a set, and \sqsubseteq is then a relation defined on A\boxtimes A.
The sets-with-preorder we obtain in this way are exactly the partial orders: sets wit... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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14c20619a2c6fa8440a773044fe9d972f70bcb81 | subsection | 69 | 110 | Order | Moreover, the -partial-order is......strong if and only if < is cotransitive: (x<z) \vdash (x<y) \vee (y<z).
...\mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}}-total if and only if (x<y)\rightarrow (x\le y) (hence < is transitive).
...\sqcup -total if and only if \le is total, (x\le y)\vee (y\le... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 942,
"openalex_id": "https://openalex.org/W2261622378",
"raw": "J. H. Conway. On numbers and games. A K Peters Ltd., Natick, MA, second edition, 2001.",
"source_ref_id": "951f95109be21f507c98556ecfdfd9e7a6992514",
"sta... | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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dd67a9a9c571ee5147310778301f192d2f607254 | subsection | 70 | 110 | Order | The coreflection of (A,\sqsubseteq ) is A with(x\circeq y) \mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}(x\sqsubseteq y) \sqcap (y\sqsubseteq x).If \sqsubseteq is a strong preorder, then \circeq is a strong equality; thus the category of strong sets is also coreflective in the category of strong preo... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.... |
4a9f68e5e745db33da142ae0ce99e9f53d7f896a | subsection | 71 | 110 | Order | And (REF ) follows by projecting from (x\circeq y) \equiv ((x\sqsubseteq y) \sqcap (y\sqsubseteq x)) to x\sqsubseteq y, from z\circeq w to z\sqsubseteq w, and applying transitivity of \sqsubseteq .Remark An affirmative preorder is automatically an âffirmative partial order on its induced set, but its induced equality i... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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c24d8552ddca626a706fdbc3988543ce724a6935 | subsection | 72 | 110 | Order | Then we have(x\sqsubset y) &\equiv (x\sqsubseteq y) \boxtimes (x\lnot \circeq y)\\
(x\sqsubseteq y) &\equiv (x\circeq y) \mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}}(x\sqsubset y).For any partial order we have(x\sqsubseteq y)\boxtimes (x\lnot \circeq y)
&\equiv (x\sqsubseteq y) \boxtimes \smash... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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276022ad133bfee3ac2a5e71f6035df2aaa0a0a8 | subsection | 73 | 110 | Real numbers and metric spaces | In this section we assume a natural numbers type permitting definitions by recursion and proof by induction.
We define addition and multiplication by recursion in the usual way, and we define a preorder on by recursion into \Omega :
(0n) def=(n+10) def=(n+1 m+1) def=(nm).
The integers are the type \times with
((a,b)... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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7f28cc3927da400ad1cf1d1d5e7e489a289b4837 | subsection | 74 | 110 | Real numbers and metric spaces | It suffices to show {\textstyle \sqcap }k. (x_n \sqsubseteq z_n + {\textstyle \frac{2}{n+1}} + {\textstyle \frac{6}{k+1}}); thus let k also be given.
Since x,z:_c, we have x_n \sqsubseteq x_k + {\textstyle \frac{1}{n+1}} + {\textstyle \frac{1}{k+1}} and z_k \sqsubseteq z_n + {\textstyle \frac{1}{n+1}} + {\textstyle \fr... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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d707ab22ecb3a1f9a28f851971d594f644acfbe2 | subsection | 75 | 110 | Real numbers and metric spaces | Given such m, for any n, |x_n-x_m| \sqsubseteq {\textstyle \frac{1}{n+1}}+{\textstyle \frac{1}{m+1}} and |y_n-y_m| \sqsubseteq {\textstyle \frac{1}{n+1}}+{\textstyle \frac{1}{m+1}}, sox_n &\,\sqsubseteq \, x_m + {\textstyle \frac{1}{n+1}}+{\textstyle \frac{1}{m+1}}
\,\sqsubseteq \, y_m + {\textstyle \frac{1}{n+1}}-{\te... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0351d6d2e475c8b1b3cd07ab9f2661cb4d6dad42 | subsection | 76 | 110 | Real numbers and metric spaces | (((s\mathrel {{\sqsubset }{\mathord {-}}}L) \boxtimes (r\sqsubset s)) \multimap (r\mathrel {{\sqsubset }{\mathord {-}}}L)).
L is upwards-open if {\textstyle \sqcap }r. ((r\mathrel {{\sqsubset }{\mathord {-}}}L) \multimap {\textstyle \bigsqcup }s. ((r\sqsubset s) \boxtimes (s\mathrel {{\sqsubset }{\mathord {-}}}L))).
... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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69e71603320bb183d11662b497001e3831e98b33 | subsection | 77 | 110 | Real numbers and metric spaces | ((r\sqsubset s) \boxtimes (s\mathrel {{\sqsubset }{\mathord {-}}}{\smash{\overline{L}}}))}\:\right.
&\qquad {\smash{\mathring{U}}}&\mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}{{\smash{\overline{L}}}}^\perp \\
{\smash{\overline{U}}}&\mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}\pro... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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ccc8ac8b5f50a1a97699754862473737e4381374 | subsection | 78 | 110 | Real numbers and metric spaces | Thus, if we write x_{\smash{\mathring{L}}},x_{\smash{\overline{L}}},x_{\smash{\mathring{U}}},x_{\smash{\overline{U}}} for the four representations of x\mathrel {{\sqsubset }{\mathord {-}}}, we have\begin{array}{ccccccccc}
(x\sqsubseteq y)&\equiv &(x_{\smash{\mathring{L}}}\sqsubseteq y_{\smash{\mathring{L}}})&\equiv &(x... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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eebbc09b2fbcaa95854db8d7633d2ad60478aec6 | subsection | 79 | 110 | Real numbers and metric spaces | If we identify r\mathrel {{\sqsubset }{\mathord {-}}} with the cut r_{\smash{\mathring{L}}}\mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}\protect \left.\:{q\mathrel {{\sqsubset }{\mathord {-}}} q \sqsubset r}\:\right., then is fully order-embedded in , and moreover for any x\mathrel {{\sqsubset }{\mat... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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56c5b5d065b3208ce8ceee98c2c62dcc88056283 | subsection | 80 | 110 | Real numbers and metric spaces | ((r<s) \wedge (r\in L)) }\hspace{28.45274pt}
L \mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}{r | \forall s. ((r<s) \rightarrow (s\in U)) }.The -set of pairs (L,U) in thm:std-ivl is also called the set of (rational) cuts , or sometimes the interval domain.
It is distinct from even classically, contain... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/s0019-3577(98)80037-2",
"end": 274,
"openalex_id": "https://openalex.org/W2074009310",
"raw": "Fred Richman. Generalized real numbers in constructive mathematics. Indagationes Mathematicae, 9(4):595 – 606, 1998.",
"source_ref_... | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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a45d6d24343e387731398a8c48b2162c83cec67b | subsection | 81 | 110 | Real numbers and metric spaces | ((x\sqsubset y) \multimap ((x\mathrel {{\sqsubset }{\mathord {-}}}L) \sqcup (y\mathrel {{\sqsubset }{\mathord {-}}}U))) and (L\sqcap U \circeq \mathord {\lnot \hspace{-2.0pt}{\scriptstyle \Box }}).
{\textstyle \sqcap }xy. (((x\lnot \mathrel {{\sqsubset }{\mathord {-}}}L) \sqcap (x\lnot \mathrel {{\sqsubset }{\mathord ... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... |
9c1dd1a8ae10ed3ae96b07ea44f01c8f3c7db5dd | subsection | 82 | 110 | Real numbers and metric spaces | Conversely, if REF and x\sqsubset y, let z = {\textstyle \frac{x+y}{2}}; then x\sqsubset z, so (x\mathrel {{\sqsubset }{\mathord {-}}}L) \sqcup (z\lnot \mathrel {{\sqsubset }{\mathord {-}}}L).
In the first case, x\mathrel {{\sqsubset }{\mathord {-}}}L, while in the second case, y\mathrel {{\sqsubset }{\mathord {-}}}U s... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.030... |
d9df95933400977b123dcaef4a2b3de510cb4615 | subsection | 83 | 110 | Real numbers and metric spaces | Since L_1 is open and x\mathrel {{\sqsubset }{\mathord {-}}}L_1, there exists y: with y\mathrel {{\sqsubset }{\mathord {-}}}L_1 and x\sqsubset y.Now x\sqsubset y implies either x\mathrel {{\sqsubset }{\mathord {-}}}L_2 or y\lnot \mathrel {{\sqsubset }{\mathord {-}}}L_2.
If x\mathrel {{\sqsubset }{\mathord {-}}}L_2, fro... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... |
ccb568bdcb01176c0f0d84a8441176e5fec0477b | subsection | 84 | 110 | Real numbers and metric spaces | If (L_1\sqsubset L_2) then there exists x: with x\mathrel {{\sqsubset }{\mathord {-}}}L_2 and x\lnot \mathrel {{\sqsubset }{\mathord {-}}}L_1.
Thus for any y:, if y\mathrel {{\sqsubset }{\mathord {-}}}L_1 we must have y\sqsubset x, since if x\sqsubseteq y we would have x\mathrel {{\sqsubset }{\mathord {-}}}L_1.
But y\s... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... |
e4034a0ed73ea26636514f3e89a0854ffbce78fb | subsection | 85 | 110 | Real numbers and metric spaces | ((q \circeq r+s) \multimap ((r \sqsubseteq x) \mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}}(s \sqsubseteq y)))\\
(q\sqsubset x \mathbin {\raisebox {1pt}{\scriptstyle }}y)
&\equiv {\textstyle \bigsqcup }q^{\prime }. ((q \sqsubset q^{\prime }) \boxtimes {\textstyle \sqcap }r s. ((q^{\prime } \circ... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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d2fde617427cfef45bee986000067b63cbd86d0b | subsection | 86 | 110 | Real numbers and metric spaces | We have shown (r \sqsubset x) \sqcup (s \sqsubset y), hence also (r \sqsubset x) \mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}}(s \sqsubset y); thus (q\sqsubset x \mathbin {\raisebox {1pt}{\scriptstyle }}y).Next suppose x,y\mathchoice{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
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ecc7d0fc1f84c0cd9afb90b63e077408d47f37a2 | subsection | 87 | 110 | Real numbers and metric spaces | ((r_k\sqsubset x) \sqcup (x\sqsubset r_{k+1})).Now we prove by induction on n that(s_{n+1}\sqsubset y) \multimap {\textstyle \bigsqcup }k.((r_k\sqsubset x) \boxtimes (s_{k+2} \sqsubset y)).The base case is n\circeq 1, in which case we can take k\mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}0.
For the ... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
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94daab0585248f6ac7219914634e1eb09581dac5 | subsection | 88 | 110 | Real numbers and metric spaces | ((q=r+s) \wedge (r<x) \wedge (s<y)).These are the standard interpretations of \mathbin {\raisebox {1pt}{\scriptstyle }} and \mathbin {\raisebox {1pt}{\scriptstyle }} respectively.If x,y\mathchoice{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\mathrel {\raisebox {-1pt}{}\mathord {\mathr... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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e13fa78e5e23c1e83fa0f0c5bead2378e9832c89 | subsection | 89 | 110 | Real numbers and metric spaces | One place where this matters is in defining metric spaces.Definition
A cut-metric on an -pre-set X is an operation d:^{X\times X} with&\vdash _{x,y\mathchoice{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\m... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
] | [
"math.LO"
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13d593b8a2e5a5b7b92d24e1f426b0922e96b0f9 | subsection | 90 | 110 | Real numbers and metric spaces | (We can also symmetrize d directly with d^{\prime }(x,y) \mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}\sup (d(x,y),d(y,x)).)
If X is already a set and d a function, the usual metric separation condition (d(x,y)\circeq 0) \vdash (x\circeq y) makes its equality coincide with that obtained in this way.I... | {
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"Michael Shulman"
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6e43ebcd2826756fecb6fae1fc02289f2588294b | subsection | 91 | 110 | Real numbers and metric spaces | But the observation of Richman is that if we treat d(a,B) as a cut, then its inequality relations to rational (hence also real) numbers are exactly what we would expect of such a “distance”.
In the standard interpretation, these become:(d(a,B) < q) &\equiv \exists b^X. ((b\in B) \wedge (d(a,b) < q))\\
(q \le d(a,B)) &... | {
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"Michael Shulman"
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8be16623809cd9e288c3b25c80a62a1c8bba095e | subsection | 92 | 110 | Real numbers and metric spaces | ((b\mathrel {{\sqsubset }{\mathord {-}}}B) \boxtimes (d(a,b) \sqsubset d(a,b^{\prime }) + \varepsilon )))whose standard interpretation, when B,B^{\prime } are affirmative, reduces to Richman's:(d(a,B) \le d(a,B^{\prime })) \equiv \forall \varepsilon . \forall \smash{b^{\prime }}^X. ((b^{\prime }\in B^{\prime }) \righta... | {
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"Michael Shulman"
] | [
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e163aef1d98a7614be7d99a35284974da64b5ff3 | subsection | 93 | 110 | Real numbers and metric spaces | ((b\mathrel {{\sqsubset }{\mathord {-}}}B) \boxtimes (d(a,b) \sqsubset r^{\prime })))\\
& {\textstyle \sqcap }b^X. ((b \mathrel {{\sqsubset }{\mathord {-}}}B) \multimap {\textstyle \bigsqcup }c^X. ((c\mathrel {{\sqsubset }{\mathord {-}}}C) \boxtimes (d(b,c) \sqsubset s^{\prime }))).Thus, for any a\mathrel {{\sqsubset }... | {
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"Michael Shulman"
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3705844129248bb64227afaaa214558774698d6f | subsection | 94 | 110 | Real numbers and metric spaces | Our solution is to replace this “additive” binary supremum with a multiplicative one: for cuts x,y we define({\textstyle \sup ^{\mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptscriptstyle \boxtimes }}}}}(x,y) \sqsubset q) &\mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}(x \sqsubset q) \boxtimes (y \s... | {
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"source_ref... | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
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d902037348e4a5017cd5fb4660654afc31a1662f | subsection | 95 | 110 | Real numbers and metric spaces | But the corresponding formulas\exists K. \forall \varepsilon . \forall nm. (n>K_\varepsilon \wedge m>K_\varepsilon \rightarrow |x_n-x_m|\le \varepsilon ) \\
\exists \varepsilon . \exists NM. \forall k. (N_k>k \wedge M_k > k \wedge |x_{N_k}-x_{M_k}|>\varepsilon ).are no longer de Morgan duals.
Gödel's “Dialectica” inter... | {
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"Michael Shulman"
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5928bf2a028def930f99ed2fb05b3b59ef0361b0 | subsection | 96 | 110 | Topology | Finally, we consider point-set topologies.
There are many classically-equivalent ways to define a topology; first we consider neighborhood relations.Note that the preorder (U\sqsubseteq V) \mathrel {\smash{\overset{\scriptscriptstyle \mathsf {def}}{=}}}\forall x^A. ((x\mathrel {{\sqsubset }{\mathord {-}}}U) \multimap (... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
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da32b283cb9bfb4770b7cacd23f048623163eae3 | subsection | 97 | 110 | Topology | ((xU) (yU))) \vdash (x\circeq y).Now a relation on A\boxtimes A is equivalently a function \mathsf {int}: A \rightarrow A, and defn:topology translates into a linear version of an “interior operator”:\mathsf {int}(U) \sqsubseteq U
(U\sqsubseteq V) \multimap (\mathsf {int}(U) \sqsubseteq \mathsf {int}(V))
\mathsf {int... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
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202a3d7770b9ad164111d4d5c1edf3c64829004f | subsection | 98 | 110 | Topology | For y\mathrel {{\sqsubset }{\mathord {-}}}U we use d(x,y) \sqsubset \varepsilon \sqsubseteq \varepsilon _U and the hypothesis from xU, and dually.
Note that here we need to use the same hypothesis d(x,y) \sqsubset \varepsilon in proving both subgoals y\mathrel {{\sqsubset }{\mathord {-}}}U and y\mathrel {{\sqsubset }{\... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
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ae5872cb2a4c9c6e8d5a429b86a375a2772e691d | subsection | 99 | 110 | Topology | ((y\in U) \rightarrow (x\ne y)).
This is the same as saying that x belongs to the inequality complement of U, i.e. x\lnot \mathrel {{\sqsubset }{\mathord {-}}}\hat{}\hspace{0.5pt}U in the standard interpretation.Theorem 10.4 Under the standard interpretation, an -topology such thatA simpler attempt at () would be (xU) ... | {
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08be6bdf6a26ae2c62a9807c3c9a67e2455f4d71 | subsection | 100 | 110 | Topology | The direct translation of -transitivity is
(x\bowtie K) \vdash (x \bowtie {y | \lnot (y\bowtie K)}),
which is equivalent to () and () together.
Our definition of equality yields(x\ne y) \equiv \exists K.((x\bowtie K) \wedge \lnot (y\bowtie K)) \vee \exists K.(\lnot (x\bowtie K) \wedge (y\bowtie K)) .However, if x\bowt... | {
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} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
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691275f8b2cb6f2a4904133dc84817044124db4d | subsection | 101 | 110 | Topology | \forall y^X.((y\in U) \rightarrow (d(x,y)\ge \varepsilon )))where the first conjunct depends only on U and the second only on U.
Thus, we may think of x \ll (U,U) as “x is in the interior of U and is apart from U”.
[Figure: The standard interpretation of an -topology]In the general case,
we can write the axioms of an ... | {
"cite_spans": []
} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
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c97d19cd47964e948a0a8664460f184cfcaf46b5 | subsection | 102 | 110 | Topology | This suggests the following definition.Definition
A unified topology on an -pre-set A consists of three predicates \ll ,\bowtie ,\mathrel {\raisebox {1pt}{\scriptstyle \delta }} on A\times \Omega ^A such that:\ll is a topology in the usual sense:
(x\ll U) &\vdash &\;& (x\in U) \\
(x\ll U) \wedge (U\subseteq V) &\vdas... | {
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} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
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6b53c539c02a9ffe85287d968860e498b961e089 | subsection | 103 | 110 | Topology | Note that transitivity for \bowtie (REF ) involves \mathrel {\raisebox {1pt}{\scriptstyle \delta }}, while binary additivity for \mathrel {\raisebox {1pt}{\scriptstyle \delta }} (REF ) (in constructively sensible form derived from \mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}}) involves \bowtie ... | {
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} | 10.1017/bsl.2022.28 | 1805.07518 | Affine logic for constructive mathematics | [
"Michael Shulman"
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"math.LO"
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cb82e574dc47c12a714439081798026b96f0d094 | subsection | 104 | 110 | Topology | A \bowtie means there is an \varepsilon >0 such that for every B\in and \varepsilon ^{\prime }<\varepsilon there is a point of A that is at least \varepsilon ^{\prime }-far from every point of B.
A \mathrel {\raisebox {1pt}{\scriptstyle \delta }} means for any \varepsilon >0 there is a B\in and an \varepsilon ^{\prime... | {
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f2c4099e9a8b49ae2ec36d7c26433f5f80bf1b38 | subsection | 105 | 110 | Towards linear constructive mathematics | So far, our primary motivation for linear logic has been parasitic on intuitionistic logic, by way of the Chu construction.
Thus, a mathematician who cares about intuitionistic logic (for any reason) may use linear logic instead, obtaining intuitionistic conclusions via the standard interpretation.
However, there are o... | {
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9516a6c29540d93d6db20cab3cc4f9c76975b2ff | subsection | 106 | 110 | Towards linear constructive mathematics | For a fixed affirmative -set A, the -subsets of A are fuzzy sets with universe A, with their usual induced metric.
Finally, (closed upper) -cuts x\mathchoice{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\mathrel {\raisebox {-1pt}{}\mathord {\mathrel {{\sqsubset }{\mathord {-}}}}}}{\mat... | {
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42a8c0c43b09a77e03f67c337a3edcd4b398a4a2 | subsection | 107 | 110 | Towards linear constructive mathematics | Thus, the “or” appearing here must be the classical one \mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}}.
That is, “if P then Q” (which we may as well start writing as P\multimap Q) is equivalent to \smash{{P}^\perp } \mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}}Q.This tells... | {
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0ba8f0bdae3ecd1ee68fb392cfe66635faefa435 | subsection | 108 | 110 | Towards linear constructive mathematics | We now let \sqcap and \boxtimes be the de Morgan duals of \sqcup and \mathbin {\raisebox {-1pt}{\rotatebox {45}{\scriptstyle \boxtimes }}}, and calculate(P \boxtimes Q) \multimap R
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1235da933c3c12f7f5571c0c5902314c22c3945a | subsection | 109 | 110 | Towards linear constructive mathematics | But this is rarely bothersome: when was the last time you saw someone prove that something is a group by assuming that it isn't and deriving a contradiction?
(See also rmk:affirm-sets,rmk:affirm-axioms.)Whether or not the reader finds the foregoing discussion convincing, I believe it proves that it is possible to argue... | {
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9069cdc2cbe359d3effd6ed604ba8cee0ddc23a3 | abstract | 0 | 115 | Abstract | The instant form and the front form of relativistic dynamics proposed by
Dirac in 1949 can be linked by an interpolation angle parameter $\delta$
spanning between the instant form dynamics (IFD) at $\delta =0$ and the front
form dynamics which is now known as the light-front dynamics (LFD) at $\delta
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4155ebcb64dae31d772bca8faf22b70c6deacb87 | subsection | 1 | 115 | Introduction | For the study of relativistic particle systems,
Dirac proposed three different forms of the relativistic
Hamiltonian dynamics in 1949: i.e. the instant (x^0 =0), front
(x^+ = (x^0 + x^3)/\sqrt{2} = 0), and point (x_\mu x^\mu = a^2 >
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e687e53a67bdaeb81a1d0f09e7bcea9978c631b1 | subsection | 2 | 115 | Introduction | In this work, we entwine the fermion propagator interpolation
with our previous works of the electromagnetic gauge field and the helicity spinors and fasten the bolts and nuts necessary to launch the interpolating QED.As we have already discussed the prototype of QED scattering processes “e\mu \rightarrow e\mu " and ... | {
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89c7ae8a573404705daa1be3792b26837f383a22 | subsection | 3 | 115 | Introduction | For the limit \delta \rightarrow 0 we have x^{\widehat{+}} = x^0
and x^{\widehat{-}} = -x^3 so that we recover usual space-time coordinates
although the z-axis is inverted while for the other extreme limit, \delta \rightarrow \frac{\pi }{4} we have x^{\widehat{\pm }} = (x^0\pm x^3)/\sqrt{2}
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c5c6ab445ebc23b6b5ff4a585c6b7a3119414b63 | subsection | 4 | 115 | Introduction | We have detailed the increment of the angle difference \theta -\theta _s with the increment of the interpolation angle \delta in Ref., which bifurcates at a critical interpolation angle \delta _{c}. We found this critical interpolation angle \delta _{c}=\arctan \left( \frac{|\mathbf {P}|}{E} \right), where |\mathbf {P}... | {
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2c3debe34612ffba3d7a5f7fd5a11f3210edf6c0 | subsection | 5 | 115 | Introduction | (REF ) and ().
In the light-front limit \delta \rightarrow \frac{\pi }{4} , i.e., \mathbb {C}\rightarrow 0 , we get\Sigma _{F,\delta \rightarrow \frac{\pi }{4}} = \frac{{q}_{on}+m}{q^2-m^2}, \hspace{21.68121pt}
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9c837244d700fa28df9b6609b09afc2b7231f7ee | subsection | 6 | 115 | Introduction | Summary and conclusions follow in Sec. . In Appendix , we derive Eq.(REF ) and present the fermion propagator in the position space which supplements the discussion
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57cd65892a04da7afd27df3608d68ed05ffba344 | subsection | 7 | 115 | Formal derivation of the Interpolation of QED | In our previous works, we studied in great detail the interpolation of the photon polarization vectors, the gauge propagator and the on-mass-shell helicity spinors.
In this paper, we complete the interpolation of the QED theory by providing the final piece of the entity: the interpolating fermion propagator.
The form o... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
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88c862731fe9b1189e286f5c27c9370c73eb4ee2 | subsection | 8 | 115 | Scattering Theory | Following what Kogut and Soper did in their light-front QED paper Although Kogut and Soper represented their work in Ref. as the QED in the infinite momentum frame, it actually was the formulation of QED in the Light-Front Dynamics (LFD)., we regard the perturbative expansion of the S matrix in Feynman diagrams as the... | {
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a8a91d857628d308afb06c4286eded778195a311 | subsection | 9 | 115 | Propagator Decomposition | In Ref. , we obtained the decomposition of the photon propagator given byD_{F}(x)_{\widehat{\mu }\widehat{\nu }}&= \int \frac{d^{2}\mathbf {q}_{\perp } }{(2\pi )^{3}}\int _{-\infty }^{\infty }d q_{\widehat{-}} \widehat{\Theta }(q_{\widehat{-}}) \frac{\mathcal {T}_{\widehat{\mu }\widehat{\nu }}}{2 \sqrt{q_{\widehat{-}}^... | {
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32de8402703165f06352c795d3c1344496694938 | subsection | 10 | 115 | Propagator Decomposition | Here, the polarization vectors \epsilon ^{\widehat{\mu }}(p,\pm ) are explicitly given in Ref. and
\mathcal {T}_{\widehat{\mu }\widehat{\nu }} given by Eq. (REF )
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6ac5b8fce498eedfd0d80b04a2a2bf84896d27ad | subsection | 11 | 115 | Propagator Decomposition | (REF ) is
the interpolating step function given by\widehat{\Theta }(q_{\widehat{-}}) &= \Theta (q_{\widehat{-}}) + (1-\delta _{\mathbb {C} 0})\Theta (-q_{\widehat{-}}) \\
&=
{\left\lbrace \begin{array}{ll}
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2904cca0cb3517bf09292d2c06f98e3c39b9c4d3 | subsection | 12 | 115 | Propagator Decomposition | \end{array}\right.}The detailed derivation of Eqs.(REF ) and (REF ) will be given in Appendix , where the pole integration is done explicitly.The result for \mathbb {C}\ne 0, i.e. \widehat{\Theta }(q_{\widehat{-}}) =1, in Eq. (REF )
can be obtained by noting the two poles for q_{\widehat{+}} in Eq. (REF ) given by{\cal... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
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"Alfredo Takashi Suzuki"
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09b66b27a726846b7972be57bf3f92e533beba87 | subsection | 13 | 115 | Propagator Decomposition | (REF ) is the invariant differential surface element on the mass shell, i.e.\int \dfrac{d^{2}\mathbf {q}_{\perp } }{(2\pi )^{3}} \dfrac{d q_{\widehat{-}}}{2 Q^{\widehat{+}}}=\int \dfrac{d^{4}q}{(2\pi )^{4}}2\pi \delta (q^{2}-m^{2}).The result for \mathbb {C} = 0, i.e. \widehat{\Theta }(q_{\widehat{-}}) = \Theta (q^+), ... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
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ad363a7a37da025607d18fec9abdec829e1cc1d1 | subsection | 14 | 115 | Propagator Decomposition | (REF ) and (REF ) to derive a decomposition for the fermion propagator
given byS_{F}(x)&=\int \dfrac{d^{2}\mathbf {q}_{\perp } }{(2\pi )^{3}}\int _{-\infty }^{\infty }d q_{\widehat{-}} \widehat{\Theta }(q_{\widehat{-}})\dfrac{1}{2 Q^{\widehat{+}}}
\left[ \Theta (x^{\widehat{+}})(q+m)e^{-i q_{\widehat{\mu }}x^{\widehat{... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
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f66c2fb47b9579153804c6c6bd8335eed292ac83 | subsection | 15 | 115 | Propagator Decomposition | This term is the instantaneous contribution unique to the LF.
Thus, when we take \mathbb {C}=0 , our fermion propagator result given by Eq. (REF ) coincides with the LF propagator previously derived by
Kogut and Soper :S_{F}(x)_{\mathrm {LF}}&=\int \dfrac{d^{2}\mathbf {q}_{\perp } }{(2\pi )^{3}}\int _{0}^{\infty } \dfr... | {
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"Chueng-Ryong Ji",
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e5a92d6a4cb2b0935523a2c5fecf379a6ac3fb18 | subsection | 16 | 115 | Rules for | To find the rules for x^{\widehat{+}}-ordered diagrams, we start with the Feynman diagrams in coordinate space.
The amplitude for diagram shown in Fig. REF for the process of e^+e^-\rightarrow \gamma \gamma can be written asi\mathcal {M}=&(-i e)^{2}\int d^{4}x d^{4}y ~\epsilon ^{*}_{\widehat{\mu }}(y) [\bar{\psi }_{2}(... | {
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"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
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0cb3a5abb8eea74e9acf710caab71d2f9618040f | subsection | 17 | 115 | Rules for | (REF ), we finish the T^{\widehat{+}} integration using the following relations\int _{-\infty }^{\infty }dT^{\widehat{+}}\Theta (T^{\widehat{+}})e^{iP_{\widehat{+}}T^{\widehat{+}}}&=\dfrac{i}{P_{\widehat{+}}+i\epsilon },\\
\int _{-\infty }^{\infty }dT^{\widehat{+}}\Theta (-T^{\widehat{+}})e^{iP_{\widehat{+}}T^{\widehat... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
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9f8400fbb740ae4659c6841a7e23a338d077e4ed | subsection | 18 | 115 | Rules for | (REF ).After the above analysis, with a little thought, one can summarize and write down the rules for x^{\widehat{+}}-ordered diagrams as the following:u(p,s), \bar{u}(p,s), v(p,s), \bar{v}(p,s), \epsilon _{\mu }(p,\lambda ), and \epsilon ^{*}_{\mu }(p,\lambda ) for each incoming and outgoing external lines;
({p}+m)=... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
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c5c0c6e22e23ce1f0698ec6434c49bb391d8da10 | subsection | 19 | 115 | Rules for | REF appearing only if \mathbb {C}=0 , i.e. only in LFD, where q^{+}=k^{\prime +}-p^{\prime +};
\frac{i}{P_{ini \widehat{+}}-P_{inter \widehat{+}}+\i \epsilon } for each internal line, where P_{int \widehat{+}} and P_{inter \widehat{+}} are the sums of energies for the initial and intermediate particles;
an over-all f... | {
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844dcb1fc00d7d36bc95823c932e29147135c004 | subsection | 20 | 115 | Equations of Motion | The Lagrangian density for QED is\mathcal {L}=-\frac{1}{4}F_{\widehat{\mu }\widehat{\nu }}F^{\widehat{\mu }\widehat{\nu }}+\bar{\psi }(i\gamma ^{\widehat{\mu }}D_{\widehat{\mu }}-m)\psi ,where D_{\widehat{\mu }}=\partial _{\widehat{\mu }}+ie A_{\widehat{\mu }}, and F_{\widehat{\mu }\widehat{\nu }}=\partial _{\widehat{\... | {
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"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
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8f04d7b7fa50e61fe04141baf0c298b4c064fb8b | subsection | 21 | 115 | Equations of Motion | We may take the boundary condition, A_{\widehat{-}}(x^{\widehat{+}},x^{1},x^{2},+\infty )=-A_{\widehat{-}}(x^{\widehat{+}},x^{1},x^{2},-\infty ), which is consistent with the choice made by Kogut and Soper for the light-front QED . Then, the solution to Eq. (REF ) is found as&A_{\widehat{-}}(x^{\widehat{+}},x^1,x^2,x^{... | {
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7343cb644b1f62782356db8fc924ffba60727964 | subsection | 22 | 115 | Equations of Motion | (REF ) becomes\bar{\nabla }^{2}A^{\widehat{+}}\equiv \left(\dfrac{\partial ^{2}}{\partial (X^{i})^{2}}+\dfrac{\partial ^{2}}{\partial (X^{\widehat{-}})^{2}}\right)A^{\widehat{+}}=-eJ^{\widehat{+}}\mathbb {C}\quad (i=1,2),which has the solutionA^{\widehat{+}}&= e \int d^{2}\mathbf {X^{\prime }}^{\perp }dX^{\prime \wideh... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
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5d112c8861ad51a97ae1c26b0d113495794dc14a | subsection | 23 | 115 | Equations of Motion | In fact, the A_{\widehat{+}} component satisfies the following constraint equation without containing any time derivatives:\bar{\nabla }^{2}(A_{\widehat{+}}+\frac{\mathbb {S} A_{\widehat{-}}}{\mathbb {C}})=(\mathbb {C}\partial _{\perp }^{2}+\partial _{\widehat{-}}^{2})(A_{\widehat{+}}+\frac{\mathbb {S} A_{\widehat{-}}}... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
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4b135ef40b06a826a9ec284dd933f4b2a1bf855a | subsection | 24 | 115 | Equations of Motion | Combining Eqs. (REF ) and (REF ), we thus get the LFD result&A^{-}(x^{+},x^{1},x^{2},x^{-})\\
=&-\frac{1}{2}\int dx^{\prime -} |x^{-}-x^{\prime -}|\left[ \partial _{-}{\partial }_{\perp }\cdot \mathbf {A}^{\perp }(x^{+},x^{1},x^{2},x^{\prime -})\right.\\ &\left.+eJ^{+}(x^{+},x^{1},x^{2},x^{\prime -})) \right],which was... | {
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fe778e036d87041f66107fa7fce184dc84ff1e8e | subsection | 25 | 115 | Equations of Motion | \\
&-\left.e\left( \gamma ^{\widehat{+}}A_{\widehat{+}}+\gamma ^{\widehat{-}}A_{\widehat{-}}+{\gamma }^{\perp }\cdot \mathbf {A}_{\perp }\right) -m\right] \psi =0,where the interpolating gamma matrices satisfy the usual Clifford algebra \lbrace \gamma ^{\widehat{\mu }},\gamma ^{\widehat{\nu }}\rbrace =2g^{\widehat{\mu ... | {
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271fbacca50bb6c4d2195cdbea72641870359530 | subsection | 26 | 115 | Equations of Motion | \\
&\left.-e\left( \gamma ^{+}A_{+}+\gamma ^{-}A_{-}+{\gamma }^{\perp }\cdot \mathbf {A}_{\perp }\right) -m\right] \psi =0,and splitting \psi into \psi _+ = P_+ \psi and \psi _- = P_- \psi with the projection operators
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2e9fc12be4a209c279cf31ffafc68bc35a8ff659 | subsection | 27 | 115 | Equations of Motion | (REF ) is obtained through the
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b44e55b16a7d174e8c0fe99ac5eb18291ce7be50 | subsection | 28 | 115 | Free Fields | The Fourier expansion of the free fermion field \psi (x) takes the form&\psi (x^{\widehat{+}},\mathbf {x}^{\perp },x^{\widehat{-}})=\int \frac{d^{2}\mathbf {p}_{\perp }dp_{\widehat{-}}}{(2\pi )^{3}(2p^{\widehat{+}})}\sum _{s=\pm 1/2} \left[ u^{(s)}e^{-ix^{\widehat{-}}p_{\widehat{-}}-i\mathbf {x}^{\perp }\cdot \mathbf {... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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92a45811835d4500138ef88198a8304f7dcdce87 | subsection | 29 | 115 | Free Fields | (REF ) and (), we find that b(\mathbf {p}_{\perp },p_{\widehat{-}};s;x^{\widehat{+}}) and d^{\dagger }(\mathbf {p}_{\perp },p_{\widehat{-}};s;x^{\widehat{+}}) satisfy the following differential equations:[i\gamma ^{\widehat{+}}\partial _{\widehat{+}}-\gamma ^{\widehat{+}}p_{\widehat{+}}]\,b(\mathbf {p}_{\perp },p_{\wid... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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60b9b6a03706236e7fcecb20f1d2635a48dd5d7e | subsection | 30 | 115 | Free Fields | \\
&\left. +v^{(s)}e^{ix^{\widehat{\mu }}p_{\widehat{\mu }}}\,d^{\dagger }(\mathbf {p}_{\perp },p_{\widehat{-}};s) \right].Following a similar procedure, we can also find the free photon field asA^{\widehat{\mu }}(x)=\int \frac{d^{2}\mathbf {p}_{\perp }dp_{\widehat{-}}}{(2\pi )^{3}2p^{\widehat{+}}}\sum _{\lambda =\pm }... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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ee23a4bda2dd20a613888943e0c420b893c52099 | subsection | 31 | 115 | Energy-Momentum and Angular Momentum Tensors | Using Noether's theorem, the conserved energy-momentum tensor and angular momentum tensor can be written as{T^{\widehat{\mu }}}_{\widehat{\nu }}&=i\bar{\psi }\gamma ^{\widehat{\mu }}\partial _{\widehat{\nu }}\psi -F^{\widehat{\mu }\widehat{\lambda }}\partial _{\widehat{\nu }}A_{\widehat{\lambda }}-{g^{\widehat{\mu }}}_... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
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dce770c64529c8796c2fcdbe53f4e32fcf53791d | subsection | 32 | 115 | Energy-Momentum and Angular Momentum Tensors | In particular, the kinematic generators which do not alter the interpolating time x^{\widehat{+}}, such as
P_{1}, P_{2}, P_{\widehat{-}}, M_{12}, M_{2\widehat{-}}, M_{1\widehat{-}}, are provided by their corresponding densities given by&{T^{\widehat{+}}}_{i}=\; i\bar{\psi }\gamma ^{\widehat{+}}\partial _{i}\psi -\parti... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
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13ed6e6d3d544bb9462488347a21b07516e98cf4 | subsection | 33 | 115 | Energy-Momentum and Angular Momentum Tensors | (REF ) and (), and thus these operators involve only independent dynamical fields \psi and A^{j} (j=1,2).Finally, the most important operator of the theory is of course the interpolating Hamiltonian density:{T^{\widehat{+}}}_{\widehat{+}}&=\bar{\psi }\left(-i\gamma ^{j}\partial _{j}-i \gamma ^{\widehat{-}}\partial _{\w... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
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659b356b158fd390208e1b86bdc697fbfbe3c5a0 | subsection | 34 | 115 | Old-fashioned Perturbation Theory | With Eqs. (REF ) - (REF ), as well as Eqs. (REF ) - (REF ), we can rewrite {T^{\widehat{+}}}_{\widehat{+}} in terms of the independent degrees of freedom A^{1}, A^{2}, \tilde{\psi }, and separate out the interaction part of the Hamiltonian density from the free part. The detailed derivation is given in Appendix .
Eq. | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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0... |
14ffea9518e2503ea859dbdee4afcb474031314a | subsection | 35 | 115 | Old-fashioned Perturbation Theory | (REF ) becomes\mathcal {H}\equiv {T^{\widehat{+}}}_{\widehat{+}}=
\mathcal {H}_{0}+\mathcal {V}with\mathcal {H}_{0}&=\bar{\tilde{\psi }}(-i\gamma ^{j}\partial _{j}-i\gamma ^{\widehat{-}}\partial _{\widehat{-}}+m)\tilde{\psi }\\
&+\dfrac{1}{4}\tilde{F}^{\widehat{\mu }\widehat{\nu }}\tilde{F}_{\widehat{\mu }\widehat{\nu ... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
"hep-th"
] | 2,018 | en | Physics | [
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581938fc3e83a4daba55fe8dfff3d26b7fdeb84b | subsection | 36 | 115 | Old-fashioned Perturbation Theory | The capital X^{\widehat{\mu }}\equiv (x^{\widehat{+}},\frac{x^1}{\sqrt{\mathbb {C}}},\frac{x^2}{\sqrt{\mathbb {C}}},x^{\widehat{-}}) is introduced previously above Eq. (REF ).
Eq. (REF ) may be considered as a generalization of Eq. (4.58) in Ref. for the quantization interpolating
between IFD and LFD.We can then calcul... | {
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} | 10.1103/PhysRevD.98.036017 | 1805.06599 | Interpolating Quantum Electrodynamics between Instant and Front Forms | [
"Chueng-Ryong Ji",
"Ziyue Li",
"Bailing Ma",
"Alfredo Takashi Suzuki"
] | [
"hep-ph",
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