Questions tagged [dimensional-analysis]
Dimensional analysis is the process of obtaining results by analysing the units and dimensions in questions, equations, and so on using The Principle of Homogeneity. Note: DO NOT USE THIS TAG if your question is about degrees of freedom or spatial dimensions.
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A dimensional analysis example in Barenblatt's book
I am reading "Scaling" by G.I. Barenblatt.
In Chapter 7 he presents an example I struggle to follow. He is looking at animals' breathing rate, and he goes:
"<..>
Our basic ...
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How total mass of universe is calculated? [closed]
I think that Total mass of universe can be calculated using below formula.
Total mass of universe = (Age of Universe) × (Planck mass / Planck time)
= (4.35×10^17 ) × (2.18×10 ^−8 / 5.39×10^−44 ) Kg
= ...
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Units of the action of a charged particle in four-potential coupling
I have a question about the definition of the action that Landau defines as:
\begin{equation}
S= -\frac{e}{c}\int A^{\mu}dx_{\mu}
\end{equation}
he says that the $1/c$ factor is introduced by ...
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Dimensional analysis of field operator for massive real scalar fields
I'm trying to make dimensional sense of the field operator resulting from the canonical quantization of the massive real scalar lagrangian:
$$\mathcal {L}=\frac 1 2 [\partial_{\mu}\phi]^2-\frac 1 2 m^...
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Why half in Lagrangian density of Klein-Gordon field? [duplicate]
The Lagrangian density for the real Klein-Gordon field, which describes a real scalar field $\phi$ with mass $m$ is given by
\begin{equation}
\mathcal{L} = \frac{1}{2} \partial^\mu \phi \, \partial_\...
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Scaling symmetry of Klein-Gordon Equation
Why does the scaling transformation \begin{align}
x^\mu &\rightarrow \lambda x^\mu \\
\end{align}
of a scalar field under the Klein-Gordon equation, \begin{equation}
\Box \phi(x) = 0
\end{equation}...
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Question on dimensions of CFT operators like $\exp({\cal O})$
I want to find the scaling dimension of the operator $e^{-iφ}$ in my system:
$$
H=H_{\text{Dot}}+H_{\text{Leads}}+H_{\text{tunneling}}+H_{\text{Env}}
$$
Where:
$$
H_{\text{Env}} = {Q^2 \over 2C}+ \...
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Units and Dimensional Analysis
I was reading a paper about "Radio emissions from pulsar companions" by F. Mottez and P. Zarka (https://arxiv.org/abs/1408.1333) and stepped over equation (16) about the relativistic ...
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Physical Interpretation of equality in 1+1D wave equation
I was re-reading Eugene Hecht's textbook on Optics, and he derived the one-dimensional differential wave equation:
$$\frac{\partial^2 \psi}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 \psi}{\...
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Doubts regarding Prahar's comments on two point correlator of free boson
In this answer @Prahar says that for a conformally invariant scalar field in $(1+1)$ dimensions, corresponding to a free boson,
$$\log|x-y|$$
is the only dimensionally consistent equation for the two-...
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How to prove that unit operators are the only operators with zero scaling dimension in an unitary CFT?
In David Simmons-Duffin's Phys 229 notes found on author's github here pg. 147 it is said that the free boson field $\phi$ in bosonic CFT has zero dimension but it is not the unit operator. So the ...
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Line integrals of vector fields other than force
Does a line integral of a vector field have any physical meaning if its function isn't a force? For instance, if I compute a line integral of an acceleration function, will this integral have work per ...
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Dirac sign flip in matter tensor
I'm reviewing P. Dirac "General Theory of Relativity" and there is some non-standard factor that I don't understand the purpose of.
In the book, at 24.4 they present the EFE we all know:
$$ ...
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Why does the energy density of a conformal field theory scale as $T^4$ in $3+1$ dimensions?
I'm trying to understand hydrodynamics of relativistic CFTs. A paper I'm referring to is this article published in PRL by Itzhak Fouxon and Yaron Oz in 2008.
The paper states that hydrodynamics ...
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Is there unit of time/space in synthetic differential geometry model in physics?
In synthetic differential geometry we deal with infinitesimals $\varepsilon$ that don't have quantities.
It's possible to apply SDG model in physics.
Simple example is function of time $f(t)=t^2$ ...
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Role of the natural temperature scale in the anomalous dimension of the renormalization group
In David Tong's lecture notes on statistical field theory, the concept of anomalous dimensions is introduced by considering the scaling of the correlation function $$\langle \phi(\mathbf{x}) \phi(\...
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The correct dimensional formula for distance travelled in $n$th second? [closed]
I have read other similar questions on this site (including The dimensional formula of distance travelled in $n$th second with the same name), but it does not specifically answer what I was looking ...
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Why is finding a mathematical basis for the fine-structure constant meaningful?
I was reading QED by Richard Feynman and at the end he mentions that:
There is a most profound and beautiful question associated with the observed coupling constant, $e$ – the amplitude for a real ...
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How to convert units when calculating a dimensionless quantity?
For instance, consider calculating this quantity:
$$
P= \frac{1}{H^4~~ } \times \left(\frac{1}{k_0}\right)^{0.2} ~~(1),
$$
where $H$ and $k_0$ are constants in units of GeV and Mpc, respectively. How ...
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Why is Perturbative expansion of gravity in terms of $GE^2$?
From General Relativity by Weinberg p.797 edited by Hawking & Israel:
This is to be used to generate a perturbation series in powers of $GE^2$ or $G/r^2$ (where $E$ and $r$ are an energy and a ...
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Planck Length Calculation
On this question, from long ago, the Planck length is calculated as the length at which the Reduced Compton Wavelength is equal to the Schwarzschild Radius. However, in the calculation, the scalar &...
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Gravitational Constant with ENM Units?
To give some context, there's a conspiracy 'theory' that I saw called Electric Universe that says that gravity is not a fundamental force and instead is a "incoherent dielectric acceleration"...
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How is Planck's constant relevant in quantum mechanics?
Planck’s constant relates the frequency of a photon to its energy but how does that relate to other quantum particles? For example in the Schrödinger equation it is used according to my book “to ...
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What is the difference between Born approximation and tree-level processes?
The answer to this question says that Born approximation is essentially equivalent to the tree-level. This can be seen from the Feynman-diagrammatic version of Born series discussed in many NRQM ...
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Where does Planck's constant come from in non-renormalizability of quantum gravity?
I am trying to understand the idea that gravity breaks down at the Planck scale, but I am confused by the use of natural units ($c = \hbar = 1$). The Einstein-Hilbert action in natural units is:
\...
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Rescaling time in differential equations
On a scientific paper, I found the following equations about a compass gait (one leg behaves like an inverted pendulum, the other one as a simple pendulum; $\theta$ and $\phi$ are time-dependent):
$$
\...
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What are natural units?
I need to make a presentation on natural units. My professor asked me to visualize a world where $c$ and $\hbar$ are actually equal to unity. Like, what are the consequences? I also want to know the ...
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Why are Critical Exponents simple non-integer powers?
I'm reading Baxter's Exactly Solved Models in Statistical Physics, and he claims that for
$$t=\frac{T-T_c}{T_c}$$
which is just a change of variable in temperature to centre and normalise w.r.t. the ...
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Convert Coulomb's law in CGS units to SI units
I recently translated the appendix to an electromagnetics text from 1945 into English. Now the client is asking me to update the formulas (I studied electrical engineering). The formulas use CGS units ...
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The Principle of Homogeneity of dimensions states that you can add,subtract quantities with same dimensions but we cannot add a constant with an angle
Both a constant and a plane and solid angle are dimensionless ie they have the same dimensions , so according to principle of homogeneity should you not be able to equate them ?
But it would be absurd ...
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What are dimensions and how are they defined? [duplicate]
We all study dimensions as a topic in physics in which we are taught the dimensions of different physical quantities but I don't understand what is the connection between the things that we study ...
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Is invariance under rescaling of the Lagrangian lost during quantization?
In classical mechanics, a field theory can be described by a lagrangian involving the field and its derivatives, $\mathcal{L}=\mathcal{L}(\phi,\,\partial\phi,\,t).$ The equations of motion for the ...
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Is there a true one-dimensional object? [closed]
I'm reviewing and expanding my knowledge of dimensions.
We live in three spatial dimensions but, apart from volume, we also have the concept of surface and curve. However, if you write a line on paper,...
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Dimensions of constants
We know that pure numbers are dimensionless then how come universal constants like the gravitational constant have a dimension cause they are also equal to some numerical value and if the numerical ...
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Fundamental and derived physical quantities
I read that fundamental physical quantities are independent of each other but, if we write length = velocity x time, then length depends on the time interval so how come it is independent?
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Why is the general unit for energy (in terms of energy bills) $\rm kWh$? [closed]
Why is the unit for energy (in terms of energy bills) expressed as $$\text{time} \cdot \frac{\text{energy}}{\text{time}}$$ rather than just energy?
Wouldn't it be better to express it in megajoules (...
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White noise fluctuation amplitude
I'm trying to understand better noise processes, and have a very basic question.
Suppose I have a stochastic process characterised by white noise, namely
$$
\langle X(t) \rangle = \overline{X} \,;\...
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Can unit prefixes be interpreted as algebraic stand-ins?
I am currently a little confused about whether unit prefixes can be interpreted algebraically
For example:
$$4km = 4\times 10^{3} m$$
Is it incorrect to say that $k$ is simply an algebraic stand in ...
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Units for the Calculus of Variations [duplicate]
Just a quick question regarding the units for a quantity. I just started reading a QFT textbook, and it starts out with a little bit of Calculus of Variations. Specifically, there is a result that ...
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How does the Planck constant enter into the uncertainty principle?
In Stein & Shakarchi's Fourier Analysis, the Fourier transform of a Schwartz function $\psi$ is defined to be
$$\hat{\psi}(\xi) = \int_{-\infty}^\infty \psi(x) e^{-2\pi i x \xi} dx$$
which gives ...
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Radian unit mystery (damped oscillator)
I would be extremely grateful for any help that anyone could offer here.
I am interested in solving the optical bloch equations for the excited state population Rabi oscillations with damping due to ...
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Extracting the dimension of an operator from algebra
I may misinterpret the question. In the lecture note of conformal field theory, arXiv:2207.09474, it says the following
where for $P^\mu=i\partial_\mu$ and $D=ix^\mu \partial_\mu$. I am confused ...
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Difference between renormalizable and super-renormalizable theories
In $\phi^n$ theory in Peskin & Schroeder the superficial degree of divergence is:
$$D = d - V[\lambda] - \big(\frac{d-2}{2}\big)N \tag{10.13}$$
where $d$ is the dimension, $V$ is the number of ...
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How to think of the unit $\rm eV$?
How to get a sense of $\rm eV$? I mean when I know how much a metre is or a second is, but how to "visualize" when it is said atomic reactions are in order of $\rm eV$ and nuclear reactions ...
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The SI-unit of the cosmological constant (vacuum energy) is $\frac{1}{m^2}$. What does that have to do with Energy?
I just don't get how Energy is measured in $\frac{1}{m^2}$. Wasn't it measured in Joules? (source is https://en.wikipedia.org/wiki/Cosmological_constant#Equation)
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How to interpret units of measurement like $\text{kg m/s}^2$?
Let's first take an example. I understand that if a car has $v=20 \,\text{m/s}$ this means that every second it moves $20 \,\text{m}$. But how should I interpret units that are multiplied like $\text{...
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If the value of Coulombs Constant is high, What can we conclude about the Electric force?
I wondered about that question and regardless of the obvious answer (If $k$ increases, the Electrostatic force increases), What can we conclude from $||\vec{F}_e||$?
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Schrodinger equation with $\hbar =1$
The Schrodinger equation is given by:
$$i \hbar \frac{d}{dt}|\psi(t)\rangle = H(t)|\psi(t)\rangle.$$
Sometimes, physicists set $\hbar=1$. I suppose that they achieve this by changing the scaling and ...
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What would be an experimental test of Sciama’s theory and why it has not been pursued yet?
Recently I came across a video were the origin of inertia was attributed to Sciama’s paper (1953).
I have seen only a couple of questions regarding this topic on Stack Physics. Both of them are ...
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Keplerian Frequency of Schwarzschild Black hole
The Keplerian frequency/ Orbital frequency is the inverse of orbital period and for Schwarzschild black hole it is given by $$\frac{1}{2\pi}\sqrt{\frac{M}{r^3}}.$$ its unit is Hertz. Now To express ...
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