Pub Date : 2022-01-12DOI: 10.1142/s0219025721500211
Anbu Arjunan
For a closed convex cone P in ℝd which is spanning and pointed, i.e. P−P=ℝd and P∩−P={0}, we consider a family of E0-semigroups over P consisting of a certain family of CCR flows and CAR flows over P and classify them up to the cocycle conjugacy.
{"title":"Remarks on CCR and CAR flows over closed convex cones","authors":"Anbu Arjunan","doi":"10.1142/s0219025721500211","DOIUrl":"https://doi.org/10.1142/s0219025721500211","url":null,"abstract":"For a closed convex cone <inline-formula><mml:math display=\"inline\" overflow=\"scroll\"><mml:mi>P</mml:mi></mml:math></inline-formula> in <inline-formula><mml:math display=\"inline\" overflow=\"scroll\"><mml:msup><mml:mrow><mml:mi>ℝ</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> which is spanning and pointed, i.e. <inline-formula><mml:math display=\"inline\" overflow=\"scroll\"><mml:mi>P</mml:mi> <mml:mo stretchy=\"false\">−</mml:mo> <mml:mi>P</mml:mi> <mml:mo>=</mml:mo> <mml:msup><mml:mrow><mml:mi>ℝ</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> and <inline-formula><mml:math display=\"inline\" overflow=\"scroll\"><mml:mi>P</mml:mi> <mml:mo stretchy=\"false\">∩</mml:mo><mml:mo stretchy=\"false\">−</mml:mo><mml:mi>P</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo><mml:mo>,</mml:mo></mml:math></inline-formula> we consider a family of <inline-formula><mml:math display=\"inline\" overflow=\"scroll\"><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>-semigroups over <inline-formula><mml:math display=\"inline\" overflow=\"scroll\"><mml:mi>P</mml:mi></mml:math></inline-formula> consisting of a certain family of CCR flows and CAR flows over <inline-formula><mml:math display=\"inline\" overflow=\"scroll\"><mml:mi>P</mml:mi></mml:math></inline-formula> and classify them up to the cocycle conjugacy.","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"19 3","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138524144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-07DOI: 10.1142/s021902572250028x
V. Crismale, S. Rossi, Paola Zurlo
Local actions of PN, the group of finite permutations on N, on quasi-local algebras are defined and proved to be PNabelian. It turns out that invariant states under local actions are automatically even, and extreme invariant states are strongly clustering. Tail algebras of invariant states are shown to obey a form of the Hewitt and Savage theorem, in that they coincide with the fixed-point von Neumann algebra. Infinite graded tensor products of C∗-algebras, which include the CAR algebra, are then addressed as particular examples of quasi-local algebras acted upon PN in a natural way. Extreme invariant states are characterized as infinite products of a single even state, and a de Finetti theorem is established. Finally, infinite products of factorial even states are shown to be factorial by applying a twisted version of the tensor product commutation theorem, which is also derived here. Mathematics Subject Classification: 46L06, 60G09, 60F20, 46L53.
{"title":"De finetti-type theorems on quasi-local algebras and infinite fermi tensor products","authors":"V. Crismale, S. Rossi, Paola Zurlo","doi":"10.1142/s021902572250028x","DOIUrl":"https://doi.org/10.1142/s021902572250028x","url":null,"abstract":"Local actions of PN, the group of finite permutations on N, on quasi-local algebras are defined and proved to be PNabelian. It turns out that invariant states under local actions are automatically even, and extreme invariant states are strongly clustering. Tail algebras of invariant states are shown to obey a form of the Hewitt and Savage theorem, in that they coincide with the fixed-point von Neumann algebra. Infinite graded tensor products of C∗-algebras, which include the CAR algebra, are then addressed as particular examples of quasi-local algebras acted upon PN in a natural way. Extreme invariant states are characterized as infinite products of a single even state, and a de Finetti theorem is established. Finally, infinite products of factorial even states are shown to be factorial by applying a twisted version of the tensor product commutation theorem, which is also derived here. Mathematics Subject Classification: 46L06, 60G09, 60F20, 46L53.","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"21 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86466299","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-29DOI: 10.1142/s0219025721500235
H. Rebei, Slaheddine Wannes
We introduce the quadratic analogue of the Bogolyubov endomorphisms of the canonical commutation relations (CCR) associated with the re-normalized square of white noise algebra (RSWN-algebra). We focus on the structure of a subclass of these endomorphisms: each of them is uniquely determined by a quadruple [Formula: see text], where [Formula: see text] are linear transformations from a test-function space [Formula: see text] into itself, while [Formula: see text] is anti-linear on [Formula: see text] and [Formula: see text] is real. Precisely, we prove that [Formula: see text] and [Formula: see text] are uniquely determined by two arbitrary complex-valued Borel functions of modulus [Formula: see text] and two maps of [Formula: see text], into itself. Under some additional analytic conditions on [Formula: see text] and [Formula: see text], we discover that we have only two equivalent classes of Bogolyubov endomorphisms, one of them corresponds to the case [Formula: see text] and the other corresponds to the case [Formula: see text]. Finally, we close the paper by building some examples in one and multi-dimensional cases.
{"title":"On the Bogolyubov endomorphisms of the renormalized square of white noise algebra","authors":"H. Rebei, Slaheddine Wannes","doi":"10.1142/s0219025721500235","DOIUrl":"https://doi.org/10.1142/s0219025721500235","url":null,"abstract":"We introduce the quadratic analogue of the Bogolyubov endomorphisms of the canonical commutation relations (CCR) associated with the re-normalized square of white noise algebra (RSWN-algebra). We focus on the structure of a subclass of these endomorphisms: each of them is uniquely determined by a quadruple [Formula: see text], where [Formula: see text] are linear transformations from a test-function space [Formula: see text] into itself, while [Formula: see text] is anti-linear on [Formula: see text] and [Formula: see text] is real. Precisely, we prove that [Formula: see text] and [Formula: see text] are uniquely determined by two arbitrary complex-valued Borel functions of modulus [Formula: see text] and two maps of [Formula: see text], into itself. Under some additional analytic conditions on [Formula: see text] and [Formula: see text], we discover that we have only two equivalent classes of Bogolyubov endomorphisms, one of them corresponds to the case [Formula: see text] and the other corresponds to the case [Formula: see text]. Finally, we close the paper by building some examples in one and multi-dimensional cases.","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"36 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89950938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-24DOI: 10.1142/s0219025721500223
N. V. Huan, N. Quang
The aim of this study is to provide some strong limit theorems for weighted sums of measurable operators. The almost uniform convergence and the bilateral almost uniform convergence are considered. As a result, we derive the strong law of large numbers for sequences of successively independent identically distributed measurable operators without using the noncommutative version of Kolmogorov’s inequality.
{"title":"Some strong limit theorems for weighted sums of measurable operators","authors":"N. V. Huan, N. Quang","doi":"10.1142/s0219025721500223","DOIUrl":"https://doi.org/10.1142/s0219025721500223","url":null,"abstract":"The aim of this study is to provide some strong limit theorems for weighted sums of measurable operators. The almost uniform convergence and the bilateral almost uniform convergence are considered. As a result, we derive the strong law of large numbers for sequences of successively independent identically distributed measurable operators without using the noncommutative version of Kolmogorov’s inequality.","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"77 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80987298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-21DOI: 10.1142/s0219025723500017
Hyun-Jung Kim, Ramiro Scorolli
We consider the (unique) mild solution $u(t,x)$ of a 1-dimensional stochastic heat equation on $[0,T]timesmathbb R$ driven by time-homogeneous white noise in the Wick-Skorokhod sense. The main result of this paper is the computation of the spatial derivative of $u(t,x)$, denoted by $partial_x u(t,x)$, and its representation as a Feynman-Kac type closed form. The chaos expansion of $partial_x u(t,x)$ makes it possible to find its (optimal) H"older regularity especially in space.
我们考虑了$[0,t]乘以mathbb R$上由时间齐次白噪声驱动的一维随机热方程的(唯一)温和解$u(t,x)$，在Wick-Skorokhod意义下。本文的主要成果是计算了$u(t,x)$的空间导数，表示为$partial_x u(t,x)$，并将其表示为Feynman-Kac型封闭形式。混乱的扩张 partial_x u (t, x)美元可以发现它(最优)H “老规律特别是在空间。
{"title":"A Feynman-Kac approach for the spatial derivative of the solution to the Wick stochastic heat equation driven by time homogeneous white noise","authors":"Hyun-Jung Kim, Ramiro Scorolli","doi":"10.1142/s0219025723500017","DOIUrl":"https://doi.org/10.1142/s0219025723500017","url":null,"abstract":"We consider the (unique) mild solution $u(t,x)$ of a 1-dimensional stochastic heat equation on $[0,T]timesmathbb R$ driven by time-homogeneous white noise in the Wick-Skorokhod sense. The main result of this paper is the computation of the spatial derivative of $u(t,x)$, denoted by $partial_x u(t,x)$, and its representation as a Feynman-Kac type closed form. The chaos expansion of $partial_x u(t,x)$ makes it possible to find its (optimal) H\"older regularity especially in space.","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"4 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88327676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-01DOI: 10.1088/978-0-7503-3655-0ch9
Jeffrey H. Williams
Kupferschmidt’s statement that “blue light is not actually blue,” expresses a conviction that is widely shared by writers, physicists, educators, and many other voices. It amounts to a definition of what color is. Blue light, and all other colored light, is said to be merely particular wavelengths. The word merely is expressive. It means that colors have no reality, that we only see things as colored, and that a colorful world is not the real world. This understanding of color denies that human experience is the basis for all knowing. Doubt about human experience is a most deeply ingrained prejudice of modern western societies, and it is their bane, estranging us from the world we live in and from ourselves. In a time when virtual reality has become a dominant part of our experience, and when questions about “fake news” and “fact checking” have become pressing, the question whether we can rely on our experiences as encounters with a real world is of heightened urgency. Centuries-long philosophical and natural scientific debate and reasoning seem to deny that the world we perceive and experience is real. However, there is a glaring inconsistency in the reasoning. People who speak like Kupferschmidt take the brain to be real. They take the instruments used for researching brain activity for real, as well as the researcher’s actions, interventions, and measurements. They take to be real everything that led to the science of electromagnetic radiation and photons with particular wavelengths. When they then come to the conclusion that brain activity is real but the perception of color is not, the reasoning becomes unreasonable. They declare one set of observations to be real, another set of observations to be unreal. When you question sense perceptions, you must also question the sense perceptions, observations, and measurements involved in brain research. The claim that denies reality to sense perceptions undermines and destroys the foundation of all natural science. If color as perceived is not real, then the brain as perceived and the measuring instruments as perceived are also not real. The late philosopher Ronald Brady argues that the statement that one class of observations (those of brain functions as observed by neuroscience) should be set apart from all other observations — like color, taste, sound, warmth, touch, balance, and so on — is not a result of experiencebased science but the result of a preference for a worldview. It is believed, but not substantiated by observation.3 Once we realize that in science we cannot shun sense experience, that the basis of all knowing is human experience, we do not ask what “is behind” and “causes” color. We do not set the class of phenomena relating to electromagnetic radiation and brain research, or any other class of phenomena, above and against the class of visual phenomena. Phenomena relate to each other, certainly, but they do not cancel each other out, the one being real, others not being real. How p
{"title":"Why is the sky blue?","authors":"Jeffrey H. Williams","doi":"10.1088/978-0-7503-3655-0ch9","DOIUrl":"https://doi.org/10.1088/978-0-7503-3655-0ch9","url":null,"abstract":"Kupferschmidt’s statement that “blue light is not actually blue,” expresses a conviction that is widely shared by writers, physicists, educators, and many other voices. It amounts to a definition of what color is. Blue light, and all other colored light, is said to be merely particular wavelengths. The word merely is expressive. It means that colors have no reality, that we only see things as colored, and that a colorful world is not the real world. This understanding of color denies that human experience is the basis for all knowing. Doubt about human experience is a most deeply ingrained prejudice of modern western societies, and it is their bane, estranging us from the world we live in and from ourselves. In a time when virtual reality has become a dominant part of our experience, and when questions about “fake news” and “fact checking” have become pressing, the question whether we can rely on our experiences as encounters with a real world is of heightened urgency. Centuries-long philosophical and natural scientific debate and reasoning seem to deny that the world we perceive and experience is real. However, there is a glaring inconsistency in the reasoning. People who speak like Kupferschmidt take the brain to be real. They take the instruments used for researching brain activity for real, as well as the researcher’s actions, interventions, and measurements. They take to be real everything that led to the science of electromagnetic radiation and photons with particular wavelengths. When they then come to the conclusion that brain activity is real but the perception of color is not, the reasoning becomes unreasonable. They declare one set of observations to be real, another set of observations to be unreal. When you question sense perceptions, you must also question the sense perceptions, observations, and measurements involved in brain research. The claim that denies reality to sense perceptions undermines and destroys the foundation of all natural science. If color as perceived is not real, then the brain as perceived and the measuring instruments as perceived are also not real. The late philosopher Ronald Brady argues that the statement that one class of observations (those of brain functions as observed by neuroscience) should be set apart from all other observations — like color, taste, sound, warmth, touch, balance, and so on — is not a result of experiencebased science but the result of a preference for a worldview. It is believed, but not substantiated by observation.3 Once we realize that in science we cannot shun sense experience, that the basis of all knowing is human experience, we do not ask what “is behind” and “causes” color. We do not set the class of phenomena relating to electromagnetic radiation and brain research, or any other class of phenomena, above and against the class of visual phenomena. Phenomena relate to each other, certainly, but they do not cancel each other out, the one being real, others not being real. How p","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"43 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84910752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-01DOI: 10.1088/978-0-7503-3655-0ch1
Jeffrey H. Williams
{"title":"The origin of units","authors":"Jeffrey H. Williams","doi":"10.1088/978-0-7503-3655-0ch1","DOIUrl":"https://doi.org/10.1088/978-0-7503-3655-0ch1","url":null,"abstract":"","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"43 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90056768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-01DOI: 10.1088/978-0-7503-3655-0ch012
Jeffrey H. Williams
{"title":"The dynamics of atoms and molecules","authors":"Jeffrey H. Williams","doi":"10.1088/978-0-7503-3655-0ch012","DOIUrl":"https://doi.org/10.1088/978-0-7503-3655-0ch012","url":null,"abstract":"","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"12 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76669139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-01DOI: 10.1088/978-0-7503-3655-0ch3
Jeffrey H. Williams
{"title":"Introduction to dimensions","authors":"Jeffrey H. Williams","doi":"10.1088/978-0-7503-3655-0ch3","DOIUrl":"https://doi.org/10.1088/978-0-7503-3655-0ch3","url":null,"abstract":"","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"3 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88804164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-01DOI: 10.1088/978-0-7503-3655-0ch7
Jeffrey H. Williams
{"title":"Rules of thumb, intuitive planning and physical insight","authors":"Jeffrey H. Williams","doi":"10.1088/978-0-7503-3655-0ch7","DOIUrl":"https://doi.org/10.1088/978-0-7503-3655-0ch7","url":null,"abstract":"","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"12 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72970380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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