What is Scalar fields: Definition and 40 Discussions
In mathematics and physics, a scalar field or scalar-valued function associates a scalar value to every point in a space – possibly physical space. The scalar may either be a (dimensionless) mathematical number or a physical quantity. In a physical context, scalar fields are required to be independent of the choice of reference frame, meaning that any two observers using the same units will agree on the value of the scalar field at the same absolute point in space (or spacetime) regardless of their respective points of origin. Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory.
Hi everyone,
In his book "Quantum field theory and the standard model", Schwartz derives the position-space Feynman rules starting from the Schwinger-Dyson formula (section 7.1.1). I have two questions about his derivation.
1) As a first step, he rewrites the correlation function as
$$...
I cannot seem to start answering the question as a result of the path not being provided. How do I solve this when the path is not provided? See picture below
Hi there. I'm trying to solve the problem mentioned above, the thing is I'm truly lost and I don't know how to start solving this problem. Sorry if I don't have a concrete attempt at a solution. How do I derive the Feynman rules for this Lagrangian? What I think happens is that in momentum...
I am looking at antenna theory and just came upon scalar fields. I found an site giving an example of a scalar field as measuring the temperature in a pan on a stove with a small layer of water. The temperature away from the heat source will be cooler than near it but it doesn't have a...
Homework Statement
Consider four real massive scalar fields, \phi_1,\phi_2,\phi_3, and \phi_4, with masses M_1,M_2,M_3,M_4.
Let these fields be coupled by the interaction lagrangian \mathcal{L}_{int}=\frac{-M_3}{2}\phi_1\phi_{3}^{2}-\frac{M_4}{2}\phi_2\phi_{4}^{2}.
Find the scattering amplitude...
I need some guidance regarding the directional derivative ...
Two books I am reading introduce the directional derivative somewhat differently ... these books are as follows:
Theodore Shifrin: Multivariable Mathematics
and
Susan Jane Colley: Vector Calculus (Second Edition)Colley...
Background and Motivation
In the Standard Model, a muon is simply an electron with a bigger mass.
But, measurements of the radius of muonic hydrogen and the muon magnetic dipole moment (muon g-2), show a fairly significant discrepancy between theory an experiment in that respect, at the five...
In the Peskin&Schröder's QFT book there's an exercise that's about a pair of scalar fields, ##\phi_1## and ##\phi_2##, having the field equations
##\left(\partial^{\mu}\partial_{\mu}+m^2 \right)\phi_1 = 0##
##\left(\partial^{\mu}\partial_{\mu}+m^2 \right)\phi_2 = 0##
where the mass parameter...
Hello! Can someone explain to me how does a scalar field changes under a Lorentz transformation? I found different notations in different places and I am a bit confused. Thank you!
Homework Statement
I really cannot seem to be able to follow the logic of how you would use the product rule when using 4 vector differential operator. ∂μ is the differential operator, Aμ is a scalar field and φ and φ* is it's complex conjugate scalar field. I have the answer, I'd just really...
The Euler-Lagrange equation obtained from the action ##S=\int\ d^{4}x\ \mathcal{L}(\phi,\partial_{\mu}\phi)## is ##\frac{\partial\mathcal{L}}{\partial\phi}-\partial_{\mu}\big(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}\big)=0##.
My goal is to generalise the Euler-Lagrange equation...
Hi. This question most probably shows my lack of understanding on the topic: why are scalar fields Lorentz invariant?
Imagine a field T(x) [x is a vector; I just don't know how to write it, sorry] that tells us the temperature in each point of a room. We make a rotation in the room and now...
In the canonical quantisation of a free scalar field ##\phi## one typical constructs a mode expansion of the corresponding field operator ##\hat{\phi}## as a solution to the Klein-Gordon equation...
This question really pertains to motivating why vectors have components whereas scalar functions do not, and why the components of a given vector transform under a coordinate transformation/ change of basis, while scalar functions transform trivially (i.e. ##\phi'(x')=\phi(x)##).
In my more...
Consider a theory with two multiplets of real scalar fields ##\phi_i## and ##\epsilon_i##, where ##i### runs
from 1 to N. The Lagrangian is given by: $$\mathcal L = \frac{1}{2} (\partial_{\mu} \phi_i) (\partial^{\mu} \phi_i) + \frac{1}{2} (\partial_{\mu} \epsilon_i) (\partial^{\mu} \epsilon_i)...
I understand that the ansatz to $$(\Box +m^{2})\phi(\mathbf{x},t)=0$$ (where ##\Box\equiv\partial^{\mu}\partial_{\mu}=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}##) is of the form ##\phi(\mathbf{x},t)=e^{(iE_{\mathbf{k}}t-\mathbf{k}\cdot\mathbf{x})}##, where...
I'm looking for a demonstration of the equation governing the evolution of the scalar field:
## \Box \phi = \frac{1}{\sqrt{g}} \frac{ \partial}{\partial x^{\mu}} \sqrt(g)g^{(\mu)(\nu)} \frac{\partial}{\partial x^{\nu}} \phi=0##
I used the lagrangian for a scalar field: ## L = \nabla_{\mu}\phi...
In the srednicki notes he goes from
$$H = \int d^{3}x a^{\dagger}(x)\left( \frac{- \nabla^{2}}{2m}\right) a(x) $$ to
$$H = \int d^{3}p\frac{1}{2m}P^{2}\tilde{a}^{\dagger}(p)\tilde{a}(p) $$
Where $$\tilde{a}(p) = \int \frac{d^{3}x}{(2\pi)^{\frac{3}{2}}}e^{-ipx}a(x)$$
Is this as simple as...
Hi. In GR , covariant differentiation is used because the basis vectors are not constant. But , what about in SR ? If the basis vectors are not Cartesian then they are not constant. Does covariant differentiation exist in SR ? And are for example spherical polar basis vectors which are not...
This is more of a QFT question, so the moderator may want to move it to another forum.
The simplest example of a QFT that I learned was the scalar field; in Sakurai's 1967 textbook.
I know the Higgs is a J=0 particle. Is it described by the simple scalar field discussed in Sakurai's text? I ask...
Hi guys,
So I've got a real scalar field which is the sum of the positive frequency part and negative frequency part:
\phi(x)=\phi^{(+)}(x)+\phi^{(-)}(y)
and I'm looking at the time-ordered product:
T(\phi(x)\phi(y))=\theta(x^{0}-y^{0})\phi(x)\phi(y)+\theta(y^{0}-x^{0})\phi(y)\phi(x)
for...
Homework Statement
I'm reading Peskin and Schroeder to the best of my ability. Other than a few integration tricks that escaped me I made it through chapter 2 with no trouble, but the beginning of chapter three, "Lorentz Invariance in Wave Equations", has me stumped. They are going through a...
If a question asks for the direction of the maximum gradient of a scalar field, is it acceptable to just use del(x) as the answer or is the question asking for a unit vector?
Thanks
After watch this video , I understood that for study the behavior of the vector field, just use 2 tools, the line integral and the surface integral, and actually too, the divergence and the curl. In accordance with this, the maxwell's equations are justly the line integral, the surface integral...
Hello! Well, I guess it's all in the title, really. I was reading about k-essence, and it was described as a scalar field having a non-canonical kinetic term. I did a bit of browsing and couldn't find a clear explanation of what, exactly, a non-canonical kinetic term is. Any help would be...
Hi,
I know this question may seem a little trivial, but is there any real difference between
\left (\partial_{\mu} \phi \right)^{\dagger} and \partial_{\mu} \phi^{\dagger}
and if so, could someone provide an explanation?
Many thanks.
(Sorry if this isn't quite in the right...
This is a really basic question, but...
Say I have a massive scalar field obeying the Klein-Gordon equation linearized about flat space,
\partial_t^2 \phi + (k^2 + m^2)\phi = 0.
This has solutions
\phi \sim e^{\pm \sqrt{k^2 + m^2}t}
and the sound speed should be
\omega_k/k =...
I know that physically, they describe relationships whereby, for instance a vector field, for each point in three dimensional space (a "vector"), we have a "vector" which has a direction or magnitude.
Now I once asked what the difference between a vector field and a vector function is and the...
So, in the calculation of D(t,r) = \left[ \phi(x) , \phi(y) \right] , where t= x^0 - y^0,~ \vec{r} = \vec{x} - \vec{y} you need to calculate the following integral
D(t,r) = \frac{1}{2\pi^2 r} \int\limits_0^\infty dp \frac{ p \sin(p r) \sin \left[(p^2 + m^2)^{1/2} t \right]} { (p^2 + m^2...
What do people mean when they say that mass renormalization of scalar field theories confronts us with a fine tuning problem. It's said the divergence in the mass of a scalar field is quadartic, rather than logarithmic, this poses a fine tuning problem. Why and how, and what does that mean...
Curl div...
Homework Statement
f is a scalar field. What does div(f) curl(f) rotgrad(f) divgrad(f) stand for?
I need to know if a scalar field can have the meanings of roration and diverge like a vector field
I'm studying the properties of the energy momentum tensor for a scalar field (linked to the electromagnetic field and corresponding energy-momentum tensor) and now I'm facing the statement:
"for a theory involving only scalar fields, the energy-momentum tensor is always symmetric". But I've...
Hi everyone,
I'm reading section 9.2 of Peskin and Schroeder, and have trouble understanding the origin of a term in the transition from equation 9.26 to 9.27. Specifically, equation 9.26 is
\frac{1}{V^2}\sum_{m,l}e^{-(k_m\cdot x_1 + k_l\cdot x_2)}\left(\prod_{k_{n}^{0}>0}\int d \Re...
Homework Statement
Working on an exercise from Srednicki's QFT and something is not clear.
Show that
[\varphi(x), M^{uv}] = \mathcal{L}^{uv} \varphi(x)
where
\mathcal{L}^{uv} = \frac{\hbar}{i} (x^u \partial^v - x^v \partial^u )
Homework Equations
(1) U(\Lambda)^{-1} \varphi(x)...
Suppose I couple a fermion field to a scalar field using \mathrm{i} g \bar{\psi}\psi \varphi and \mathrm{i} g \bar{\psi}\gamma_5\psi\varphi.
I'm trying to understand what would be the physical difference between these interactions. I know that (1/2)(1\pm \gamma_5) approximately projects out...
Hi,
I'm trying to show that electromagnetism and scalar field theories satisfy the DEC. I know how to find T_{\mu\nu} and all that and what I have to show (T_{\mu\nu} T^\nu_{\ \lambda} t^\mu t^\lambda\leq 0 and T_{\mu\nu} t^\mu t^\nu\geq 0 for timelike t^\mu), but I'm having trouble getting...
I'm lazy so I'm going to start bringing my questions here.
Correct me if I'm wrong, but isn't it true that baryons only enter the standard model through this subtle topological effect. Now this is where I'm at. I kind of got the Goldstone boson concept, maybe someone could better explain...
"...without resorting to scalar fields"
http://arxiv.org/astro-ph/0703566
Co-authored by Parampreet Singh, one of the experts in Quantum Cosmology (gauged by publication trackrecord and citations by other scholars, see:
https://www.physicsforums.com/showthread.php?p=1368143#post1368143 )...
Dear PF,
Can I consider the composite field for instance psi_bar psi as a scalar filed?
I mean can it be the same in all respects? Can this composite field and scalar filed treated as totally equivalent?
Thks