How this definition of functional integral leads to the Wiener measure?
$begingroup$
In Quantum Mechanics and Quantum Field Theory texts, one encounters a definition of functional integral in terms of a limiting process associated to a time-slicing procedure. This is the way I've found best to summarize what is done in more mathematically precise terms:
Definition: Let $[a,b]subset mathbb{R}$ and let a partition $P$ of $[a,b]$ be given: $$a=t_0<t_1<cdots<t_{N-1}<t_N=b.$$ Let $x_0,dots, x_{N}in mathbb{R}$. We call the piecewise smooth path $x : [a,b]to mathbb{R}$ given by $$x(t)=x_k+frac{x_{k+1}-x_k}{t_{k+1}-t_k}(t-t_k),quad tin [t_{k},t_{k+1}],quad kin {0,dots, N}$$
a linear interpolation of the points $x_0,dots, x_N$.
Definition: Let $C_p^{infty}([a,b];mathbb{R})$ be the space of piecewise smooth paths defined in $[a,b]$. Let further $mathfrak{F}: C_p^infty([a,b];mathbb{R})to mathbb{C}$ be a given functional acting on such paths.
Let further $P$ be a partition of $[a,b]$ into $N$ subintervals. We define the functional $mathfrak{F}_P : mathbb{R}^{N+1}to mathbb{C}$ with respect to this partition to be
$$mathfrak{F}_P(x_0,dots,x_N)=mathfrak{F}[x_P(t)]$$
where $x_P(t)$ is the linear interpolation of the points $x_0,dots, x_N$.
Definition: Let $C^infty_p([a,b];mathbb{R})$ be the space of piecewise smooth paths defined in $[a,b]$. Let a functional $mathfrak{F}: C^infty_p([a,b];mathbb{R})to mathbb{C}$ be given.
We say that $mathfrak{F}$ is functionally integrable if the limit:
$$lim_{|P|to 0}int_{mathbb{R}^{N+1}} mathfrak{F}_P(x_0,dots, x_N) dx_0cdots dx_{N}$$
exists, where $|P|$ is the mesh of the partition, given by $$|P|=max{t_{k+1}-t_k | kin {0,dots, N}}.$$ In that case we call such limit the functional integral of $mathfrak{F}$ and denote it by
$$int mathcal{D}x(t) mathfrak{F}[x(t)].$$
So again, this process defines the symbol
$$int_{C_p^{infty}([a,b];mathbb{R})}mathfrak{F}[x(t)]mathcal{D}x(t)$$
for "functionally integrable" functionals. This is not at first a Lebesgue integral with respect to some measure on the space of paths.
I've heard a few times though, that this integral so defined, does indeed come from a measure and that it is the Wiener measure. So if I got it right, in a sense, one can show that this integral actually defines the Wiener measure and is the corresponding Lebesgue integral.
How can that be shown? How this definition of functional integral yields the Wiener measure and how this integral defined as a limit is the Lebesgue integral with respect to said measure?
integration functional-analysis measure-theory mathematical-physics quantum-mechanics
asked Dec 12 '18 at 11:43
user1620696user1620696
11.6k442115
$endgroup$
add a comment |
$begingroup$
In Quantum Mechanics and Quantum Field Theory texts, one encounters a definition of functional integral in terms of a limiting process associated to a time-slicing procedure. This is the way I've found best to summarize what is done in more mathematically precise terms:
Definition: Let $[a,b]subset mathbb{R}$ and let a partition $P$ of $[a,b]$ be given: $$a=t_0<t_1<cdots<t_{N-1}<t_N=b.$$ Let $x_0,dots, x_{N}in mathbb{R}$. We call the piecewise smooth path $x : [a,b]to mathbb{R}$ given by $$x(t)=x_k+frac{x_{k+1}-x_k}{t_{k+1}-t_k}(t-t_k),quad tin [t_{k},t_{k+1}],quad kin {0,dots, N}$$
a linear interpolation of the points $x_0,dots, x_N$.
Definition: Let $C_p^{infty}([a,b];mathbb{R})$ be the space of piecewise smooth paths defined in $[a,b]$. Let further $mathfrak{F}: C_p^infty([a,b];mathbb{R})to mathbb{C}$ be a given functional acting on such paths.
Let further $P$ be a partition of $[a,b]$ into $N$ subintervals. We define the functional $mathfrak{F}_P : mathbb{R}^{N+1}to mathbb{C}$ with respect to this partition to be
$$mathfrak{F}_P(x_0,dots,x_N)=mathfrak{F}[x_P(t)]$$
where $x_P(t)$ is the linear interpolation of the points $x_0,dots, x_N$.
Definition: Let $C^infty_p([a,b];mathbb{R})$ be the space of piecewise smooth paths defined in $[a,b]$. Let a functional $mathfrak{F}: C^infty_p([a,b];mathbb{R})to mathbb{C}$ be given.
We say that $mathfrak{F}$ is functionally integrable if the limit:
$$lim_{|P|to 0}int_{mathbb{R}^{N+1}} mathfrak{F}_P(x_0,dots, x_N) dx_0cdots dx_{N}$$
exists, where $|P|$ is the mesh of the partition, given by $$|P|=max{t_{k+1}-t_k | kin {0,dots, N}}.$$ In that case we call such limit the functional integral of $mathfrak{F}$ and denote it by
$$int mathcal{D}x(t) mathfrak{F}[x(t)].$$
So again, this process defines the symbol
$$int_{C_p^{infty}([a,b];mathbb{R})}mathfrak{F}[x(t)]mathcal{D}x(t)$$
for "functionally integrable" functionals. This is not at first a Lebesgue integral with respect to some measure on the space of paths.
I've heard a few times though, that this integral so defined, does indeed come from a measure and that it is the Wiener measure. So if I got it right, in a sense, one can show that this integral actually defines the Wiener measure and is the corresponding Lebesgue integral.
How can that be shown? How this definition of functional integral yields the Wiener measure and how this integral defined as a limit is the Lebesgue integral with respect to said measure?
integration functional-analysis measure-theory mathematical-physics quantum-mechanics
asked Dec 12 '18 at 11:43
user1620696user1620696
11.6k442115
$endgroup$
add a comment |
$begingroup$
In Quantum Mechanics and Quantum Field Theory texts, one encounters a definition of functional integral in terms of a limiting process associated to a time-slicing procedure. This is the way I've found best to summarize what is done in more mathematically precise terms:
Definition: Let $[a,b]subset mathbb{R}$ and let a partition $P$ of $[a,b]$ be given: $$a=t_0<t_1<cdots<t_{N-1}<t_N=b.$$ Let $x_0,dots, x_{N}in mathbb{R}$. We call the piecewise smooth path $x : [a,b]to mathbb{R}$ given by $$x(t)=x_k+frac{x_{k+1}-x_k}{t_{k+1}-t_k}(t-t_k),quad tin [t_{k},t_{k+1}],quad kin {0,dots, N}$$
a linear interpolation of the points $x_0,dots, x_N$.
Definition: Let $C_p^{infty}([a,b];mathbb{R})$ be the space of piecewise smooth paths defined in $[a,b]$. Let further $mathfrak{F}: C_p^infty([a,b];mathbb{R})to mathbb{C}$ be a given functional acting on such paths.
Let further $P$ be a partition of $[a,b]$ into $N$ subintervals. We define the functional $mathfrak{F}_P : mathbb{R}^{N+1}to mathbb{C}$ with respect to this partition to be
$$mathfrak{F}_P(x_0,dots,x_N)=mathfrak{F}[x_P(t)]$$
where $x_P(t)$ is the linear interpolation of the points $x_0,dots, x_N$.
Definition: Let $C^infty_p([a,b];mathbb{R})$ be the space of piecewise smooth paths defined in $[a,b]$. Let a functional $mathfrak{F}: C^infty_p([a,b];mathbb{R})to mathbb{C}$ be given.
We say that $mathfrak{F}$ is functionally integrable if the limit:
$$lim_{|P|to 0}int_{mathbb{R}^{N+1}} mathfrak{F}_P(x_0,dots, x_N) dx_0cdots dx_{N}$$
exists, where $|P|$ is the mesh of the partition, given by $$|P|=max{t_{k+1}-t_k | kin {0,dots, N}}.$$ In that case we call such limit the functional integral of $mathfrak{F}$ and denote it by
$$int mathcal{D}x(t) mathfrak{F}[x(t)].$$
So again, this process defines the symbol
$$int_{C_p^{infty}([a,b];mathbb{R})}mathfrak{F}[x(t)]mathcal{D}x(t)$$
for "functionally integrable" functionals. This is not at first a Lebesgue integral with respect to some measure on the space of paths.
I've heard a few times though, that this integral so defined, does indeed come from a measure and that it is the Wiener measure. So if I got it right, in a sense, one can show that this integral actually defines the Wiener measure and is the corresponding Lebesgue integral.
How can that be shown? How this definition of functional integral yields the Wiener measure and how this integral defined as a limit is the Lebesgue integral with respect to said measure?
integration functional-analysis measure-theory mathematical-physics quantum-mechanics
asked Dec 12 '18 at 11:43
user1620696user1620696
11.6k442115
$endgroup$
In Quantum Mechanics and Quantum Field Theory texts, one encounters a definition of functional integral in terms of a limiting process associated to a time-slicing procedure. This is the way I've found best to summarize what is done in more mathematically precise terms:
Definition: Let $[a,b]subset mathbb{R}$ and let a partition $P$ of $[a,b]$ be given: $$a=t_0<t_1<cdots<t_{N-1}<t_N=b.$$ Let $x_0,dots, x_{N}in mathbb{R}$. We call the piecewise smooth path $x : [a,b]to mathbb{R}$ given by $$x(t)=x_k+frac{x_{k+1}-x_k}{t_{k+1}-t_k}(t-t_k),quad tin [t_{k},t_{k+1}],quad kin {0,dots, N}$$
a linear interpolation of the points $x_0,dots, x_N$.
Definition: Let $C_p^{infty}([a,b];mathbb{R})$ be the space of piecewise smooth paths defined in $[a,b]$. Let further $mathfrak{F}: C_p^infty([a,b];mathbb{R})to mathbb{C}$ be a given functional acting on such paths.
Let further $P$ be a partition of $[a,b]$ into $N$ subintervals. We define the functional $mathfrak{F}_P : mathbb{R}^{N+1}to mathbb{C}$ with respect to this partition to be
$$mathfrak{F}_P(x_0,dots,x_N)=mathfrak{F}[x_P(t)]$$
where $x_P(t)$ is the linear interpolation of the points $x_0,dots, x_N$.
Definition: Let $C^infty_p([a,b];mathbb{R})$ be the space of piecewise smooth paths defined in $[a,b]$. Let a functional $mathfrak{F}: C^infty_p([a,b];mathbb{R})to mathbb{C}$ be given.
We say that $mathfrak{F}$ is functionally integrable if the limit:
$$lim_{|P|to 0}int_{mathbb{R}^{N+1}} mathfrak{F}_P(x_0,dots, x_N) dx_0cdots dx_{N}$$
exists, where $|P|$ is the mesh of the partition, given by $$|P|=max{t_{k+1}-t_k | kin {0,dots, N}}.$$ In that case we call such limit the functional integral of $mathfrak{F}$ and denote it by
$$int mathcal{D}x(t) mathfrak{F}[x(t)].$$
So again, this process defines the symbol
$$int_{C_p^{infty}([a,b];mathbb{R})}mathfrak{F}[x(t)]mathcal{D}x(t)$$
for "functionally integrable" functionals. This is not at first a Lebesgue integral with respect to some measure on the space of paths.
I've heard a few times though, that this integral so defined, does indeed come from a measure and that it is the Wiener measure. So if I got it right, in a sense, one can show that this integral actually defines the Wiener measure and is the corresponding Lebesgue integral.
How can that be shown? How this definition of functional integral yields the Wiener measure and how this integral defined as a limit is the Lebesgue integral with respect to said measure?
integration functional-analysis measure-theory mathematical-physics quantum-mechanics
integration functional-analysis measure-theory mathematical-physics quantum-mechanics
asked Dec 12 '18 at 11:43
user1620696user1620696
11.6k442115
asked Dec 12 '18 at 11:43
user1620696user1620696
11.6k442115
asked Dec 12 '18 at 11:43
user1620696user1620696
11.6k442115
asked Dec 12 '18 at 11:43
user1620696user1620696
11.6k442115
asked Dec 12 '18 at 11:43
user1620696user1620696
11.6k442115
11.6k442115
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3036583%2fhow-this-definition-of-functional-integral-leads-to-the-wiener-measure%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3036583%2fhow-this-definition-of-functional-integral-leads-to-the-wiener-measure%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
