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I've been reading about the renormalization group in QFT from Peskin & Schroeder, and wanted to consolidate understanding of "irrelevant operator" by connecting it to something more intuitive, aka Brownian motion. I'd be particularly interested in references exploring analogies between stochastic processes and (statistical or quantum) field theory, and the renormalization group.

My understanding of Brownian motion, is that the "IR" or "coarse grained theory" is well described by a diffusion equation:

$$ {\partial \over \partial t} f(x, t) - D {\partial^2 \over \partial^2 x} f(x, t) = 0$$

where $f(x,t)$ would represent some probability density function. The constant D is some finite value "coupling constant", measured in the IR in lab, and perfectly models the IR physics.

Random walk as a model for Brownian Motion However, if we were to write

Here's a discrete time model for brownian motion, we could model it asthe above diffusion equation:

$$ x(t_{n+1}) = v \times n(t_n) \Delta t + x(t_n)$$

$v$ is the UV "coupling" and $n(t_n)$ is a set of IID random walk with time steps $\Delta t$ with some "diffusion coefficient" modelled by the probability to move left or right with some velocityvariables. As

In the model above, as we go deep into the UV, the velocity to correctly model a finite $D$ in the IR scales as ${D \over \sqrt{\Delta t}}$, because of the central limit theorem. We therefore know that our model of physics cannot be correct to arbitrary time scales, hence the velocity generating brownian motion in the IR (or coupling constant) being an "irrelevant operator." 

Question(s)

Is this model an example of the renormalization group at work?
If yes, what is the irrelevant operator? How does the degree of superficial divergence show up? (my gut says the dimensionality of the velocity in the discrete time model). What kind of "loop corrections" to the RG flow can we incorporate in such a simple model of physics at different time scales (aka anomalous dimensions)?

Comment & Discussion

I am aware the random walk has a path integral solution in the continuum. If I generalize the random walk of a point particle to a random of a field (the observable now being a statistical field in space/time rather than a coordinate). How does this relate to the renormalization group?

Thank you!

I've been reading about the renormalization group in QFT from Peskin & Schroeder, and wanted to consolidate understanding of "irrelevant operator" by connecting it to something more intuitive, aka Brownian motion. I'd be particularly interested in references exploring analogies between stochastic processes and (statistical or quantum) field theory, and the renormalization group.

My understanding of Brownian motion, is that the "IR" or "coarse grained theory" is well described by a diffusion equation:

$$ {\partial \over \partial t} f(x, t) - D {\partial^2 \over \partial^2 x} f(x, t) = 0$$

where $f(x,t)$ would represent some probability density function. The constant D is some finite value "coupling constant", measured in the IR in lab, and perfectly models the IR physics.

Random walk as a model for Brownian Motion However, if we were to write a discrete time model for brownian motion, we could model it as a random walk with time steps $\Delta t$ with some "diffusion coefficient" modelled by the probability to move left or right with some velocity. As we go deep into the UV, the velocity to correctly model a finite $D$ in the IR scales as ${D \over \sqrt{\Delta t}}$, because of the central limit theorem. We therefore know that our model of physics cannot be correct to arbitrary time scales, hence the velocity generating brownian motion in the IR (or coupling constant) being an "irrelevant operator."

Question(s)

Is this model an example of the renormalization group at work?
If yes, what is the irrelevant operator? How does the degree of superficial divergence show up? (my gut says the dimensionality of the velocity in the discrete time model). What kind of "loop corrections" to the RG flow can we incorporate in such a simple model of physics at different time scales (aka anomalous dimensions)?

Comment & Discussion

I am aware the random walk has a path integral solution in the continuum. If I generalize the random walk of a point particle to a random of a field (the observable now being a statistical field in space/time rather than a coordinate). How does this relate to the renormalization group?

Thank you!

I've been reading about the renormalization group in QFT from Peskin & Schroeder, and wanted to consolidate understanding of "irrelevant operator" by connecting it to something more intuitive, aka Brownian motion. I'd be particularly interested in references exploring analogies between stochastic processes and (statistical or quantum) field theory, and the renormalization group.

My understanding of Brownian motion, is that the "IR" or "coarse grained theory" is well described by a diffusion equation:

$$ {\partial \over \partial t} f(x, t) - D {\partial^2 \over \partial^2 x} f(x, t) = 0$$

where $f(x,t)$ would represent some probability density function. The constant D is some finite value "coupling constant", measured in the IR in lab, and perfectly models the IR physics.

Random walk as a model for Brownian Motion

Here's a discrete time model for the above diffusion equation:

$$ x(t_{n+1}) = v \times n(t_n) \Delta t + x(t_n)$$

$v$ is the UV "coupling" and $n(t_n)$ is a set of IID random variables.

In the model above, as we go deep into the UV, the velocity to correctly model a finite $D$ in the IR scales as ${D \over \sqrt{\Delta t}}$, because of the central limit theorem. We therefore know that our model of physics cannot be correct to arbitrary time scales, hence the velocity generating brownian motion in the IR (or coupling constant) being an "irrelevant operator." 

Question(s)

Is this model an example of the renormalization group at work?
If yes, what is the irrelevant operator? How does the degree of superficial divergence show up? (my gut says the dimensionality of the velocity in the discrete time model). What kind of "loop corrections" to the RG flow can we incorporate in such a simple model of physics at different time scales (aka anomalous dimensions)?

Comment & Discussion

I am aware the random walk has a path integral solution in the continuum. If I generalize the random walk of a point particle to a random of a field (the observable now being a statistical field in space/time rather than a coordinate). How does this relate to the renormalization group?

Thank you!

3 deleted 3 characters in body
source | link

I've been reading about the renormalization group in QFT from Peskin & Schroeder, and wanted to consolidate understanding of "irrelevant operator" by connecting it to something more intuitive, aka Brownian motion. I'd be particularly interested in references exploring analogies between stochastic processes and (statistical or quantum) field theory, and the renormalization group.

My understanding of Brownian motion, is that the "IR" or "coarse grained theory" is well described by a diffusion equation:

$$ {\partial \over \partial t} f(x, t) - D {\partial^2 \over \partial^2 x} f(x, t) = 0$$

where $f(x,t)$ would represent some probability density function. The constant D is some finite value "coupling constant", measured in the IR in lab, and perfectly models the IR physics.

Random walk as a model for Brownian Motion However, if we were to write a discrete time model for brownian motion, we could model it as a random walk with time steps $\Delta t$ with some "diffusion coefficient" modelled by the probability to move left or right with some velocity. As we go deep into the UV, the velocity to correctly model a finite $D$ in the IR scales as ${D \over \sqrt{\Delta t}}$, because of the central limit theorem. We therefore know that our model of physics cannot be correct to arbitrary time scales, hence the velocity generating brownian motion in the IR (or coupling constant) being an "irrelevant operator."

Question(s)

Is this model an example of the renormalization group at work?
If yes, what is the irrelevant operator? How does the degree of superficial divergence show up? (my gut says the dimensionality of the velocity in the discrete time model). What kind of "loop corrections" to the RG flow can we incorporate in such a simple model of physics at different time scales (aka anomalous dimensions)?

Comment & Discussion

I am aware the random walk ashas a path integral solution in the continuum. If I generalize the random walk of a point particle to a random of a field (the observable now being a statistical stochastic field in space/time rather than a positioncoordinate). How does this relate to the renormalization group flow of coupling constants in statistical field theories?

Thank you!

I've been reading about the renormalization group in QFT from Peskin & Schroeder, and wanted to consolidate understanding of "irrelevant operator" by connecting it to something more intuitive, aka Brownian motion. I'd be particularly interested in references exploring analogies between stochastic processes and (statistical or quantum) field theory, and the renormalization group.

My understanding of Brownian motion, is that the "IR" or "coarse grained theory" is well described by a diffusion equation:

$$ {\partial \over \partial t} f(x, t) - D {\partial^2 \over \partial^2 x} f(x, t) = 0$$

where $f(x,t)$ would represent some probability density function. The constant D is some finite value "coupling constant", measured in the IR in lab, and perfectly models the IR physics.

Random walk as a model for Brownian Motion However, if we were to write a discrete time model for brownian motion, we could model it as a random walk with time steps $\Delta t$ with some "diffusion coefficient" modelled by the probability to move left or right with some velocity. As we go deep into the UV, the velocity to correctly model a finite $D$ in the IR scales as ${D \over \sqrt{\Delta t}}$, because of the central limit theorem. We therefore know that our model of physics cannot be correct to arbitrary time scales, hence the velocity generating brownian motion in the IR (or coupling constant) being an "irrelevant operator."

Question(s)

Is this model an example of the renormalization group at work?
If yes, what is the irrelevant operator? How does the degree of superficial divergence show up? (my gut says the dimensionality of the velocity in the discrete time model). What kind of "loop corrections" to the RG flow can we incorporate in such a simple model of physics at different time scales (aka anomalous dimensions)?

Comment & Discussion

I am aware the random walk as a path integral solution in the continuum. If I generalize the random walk of a point particle to a random of a field (the observable now being a statistical stochastic field rather than a position). How does this relate to the renormalization group flow of coupling constants in statistical field theories?

Thank you!

I've been reading about the renormalization group in QFT from Peskin & Schroeder, and wanted to consolidate understanding of "irrelevant operator" by connecting it to something more intuitive, aka Brownian motion. I'd be particularly interested in references exploring analogies between stochastic processes and (statistical or quantum) field theory, and the renormalization group.

My understanding of Brownian motion, is that the "IR" or "coarse grained theory" is well described by a diffusion equation:

$$ {\partial \over \partial t} f(x, t) - D {\partial^2 \over \partial^2 x} f(x, t) = 0$$

where $f(x,t)$ would represent some probability density function. The constant D is some finite value "coupling constant", measured in the IR in lab, and perfectly models the IR physics.

Random walk as a model for Brownian Motion However, if we were to write a discrete time model for brownian motion, we could model it as a random walk with time steps $\Delta t$ with some "diffusion coefficient" modelled by the probability to move left or right with some velocity. As we go deep into the UV, the velocity to correctly model a finite $D$ in the IR scales as ${D \over \sqrt{\Delta t}}$, because of the central limit theorem. We therefore know that our model of physics cannot be correct to arbitrary time scales, hence the velocity generating brownian motion in the IR (or coupling constant) being an "irrelevant operator."

Question(s)

Is this model an example of the renormalization group at work?
If yes, what is the irrelevant operator? How does the degree of superficial divergence show up? (my gut says the dimensionality of the velocity in the discrete time model). What kind of "loop corrections" to the RG flow can we incorporate in such a simple model of physics at different time scales (aka anomalous dimensions)?

Comment & Discussion

I am aware the random walk has a path integral solution in the continuum. If I generalize the random walk of a point particle to a random of a field (the observable now being a statistical field in space/time rather than a coordinate). How does this relate to the renormalization group?

Thank you!

2 deleted 3 characters in body
source | link

I've been reading some about the renormalization group in QFT from Peskin & Schroeder, and wanted to consolidate understanding of "irrelevant operator" by connecting it to something more intuitive, aka Brownian motion. I'd be particularly interested in references exploring analogies between stochastic processes and (statistical or quantum) field theory, and the renormalization group.

My understanding of Brownian motion, is that the "IR" or "coarse grained theory" is well described by a diffusion equation:

$$ {\partial \over \partial t} f(x, t) - D {\partial^2 \over \partial^2 x} f(x, t) = 0$$

where $f(x,t)$ would represent some probability density function. The constant D is some finite value "coupling constant", measured in the IR in lab, and perfectly models the IR physics.

Random walk as a model for Brownian Motion However, if we were to write a discrete space-timetime model for brownian motion, we could model it as a random walk with time steps $\Delta t$ with some "diffusion coefficient" modelled by the probability to move left or right with some velocity. As we go deep into the UV, the velocity to correctly model a finite $D$ in the IR scales as ${D \over \sqrt{\Delta t}}$, because of the central limit theorem. We therefore know that our model of physics cannot be correct to arbitrary length/timetime scales, hence the velocity generating brownian motion in the IR (or coupling constant) being an "irrelevant operator."

Question IsQuestion(s)

Is this model an example of the renormalization group at work?
If yes, what is the irrelevant operator? How does the degree of superficial divergence show up? (my gut says the dimensionality of the velocity in the discrete time model). What kind of "loop corrections" to the RG flow can we incorporate in such a simple model of physics at different time scales (aka anomalous dimensions)?

Further comment IComment & Discussion

I am aware the random walk as a path integral solution in the continuum. If I generalize the random walk of a point particle to a random of a field (the observable now being a statistical stochastic field rather than a position). How does this relate to the renormalization group flow of coupling constants in statistical field theories?

Thank you!

I've been reading some about the renormalization group in QFT, and wanted to consolidate understanding of "irrelevant operator" by connecting it to something more intuitive, aka Brownian motion. I'd be particularly interested in references exploring analogies between stochastic processes and (statistical) field theory, and the renormalization group.

My understanding of Brownian motion, is that the "IR" or "coarse grained theory" is well described by a diffusion equation:

$$ {\partial \over \partial t} f(x, t) - D {\partial^2 \over \partial^2 x} f(x, t) = 0$$

where $f(x,t)$ would represent some probability density function. The constant D is some finite value "coupling constant", measured in the IR in lab, and perfectly models the IR physics.

Random walk as a model for Brownian Motion However, if we were to write a discrete space-time model for brownian motion, we could model it as a random walk with time steps $\Delta t$ with some "diffusion coefficient" modelled by the probability to move left or right with some velocity. As we go deep into the UV, the velocity to correctly model a finite $D$ in the IR scales as ${D \over \sqrt{\Delta t}}$, because of the central limit theorem. We therefore know that our model of physics cannot be correct to arbitrary length/time scales, hence the velocity generating brownian motion (or coupling constant) being an "irrelevant operator."

Question Is this model an example of the renormalization group at work?
If yes, what is the irrelevant operator? How does the degree of superficial divergence show up? (my gut says the dimensionality of the velocity in the discrete time model). What kind of "loop corrections" to the RG flow can we incorporate in such a simple model of physics at different time scales (aka anomalous dimensions)?

Further comment I am aware the random walk as a path integral solution in the continuum. If I generalize the random walk of a point particle to a random of a field (the observable now being a statistical stochastic field rather than a position). How does this relate to the renormalization group flow of coupling constants in statistical field theories?

Thank you!

I've been reading about the renormalization group in QFT from Peskin & Schroeder, and wanted to consolidate understanding of "irrelevant operator" by connecting it to something more intuitive, aka Brownian motion. I'd be particularly interested in references exploring analogies between stochastic processes and (statistical or quantum) field theory, and the renormalization group.

My understanding of Brownian motion, is that the "IR" or "coarse grained theory" is well described by a diffusion equation:

$$ {\partial \over \partial t} f(x, t) - D {\partial^2 \over \partial^2 x} f(x, t) = 0$$

where $f(x,t)$ would represent some probability density function. The constant D is some finite value "coupling constant", measured in the IR in lab, and perfectly models the IR physics.

Random walk as a model for Brownian Motion However, if we were to write a discrete time model for brownian motion, we could model it as a random walk with time steps $\Delta t$ with some "diffusion coefficient" modelled by the probability to move left or right with some velocity. As we go deep into the UV, the velocity to correctly model a finite $D$ in the IR scales as ${D \over \sqrt{\Delta t}}$, because of the central limit theorem. We therefore know that our model of physics cannot be correct to arbitrary time scales, hence the velocity generating brownian motion in the IR (or coupling constant) being an "irrelevant operator."

Question(s)

Is this model an example of the renormalization group at work?
If yes, what is the irrelevant operator? How does the degree of superficial divergence show up? (my gut says the dimensionality of the velocity in the discrete time model). What kind of "loop corrections" to the RG flow can we incorporate in such a simple model of physics at different time scales (aka anomalous dimensions)?

Comment & Discussion

I am aware the random walk as a path integral solution in the continuum. If I generalize the random walk of a point particle to a random of a field (the observable now being a statistical stochastic field rather than a position). How does this relate to the renormalization group flow of coupling constants in statistical field theories?

Thank you!

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