Scalar Definition and 777 Threads

Hi. General question: Is there a fixed way to find all invariant tensor for a generic representation? Example problem: Suppose you search for all indipendent quartic interactions of a scalar octet field ## \phi^{a} ## in the adjoint representation of SU(3). They will be terms like ##...

1. Does anyone know why for an isotropic distribution function, pressure tensor reduces to a scalar pressure? For instance, for a Maxwellian distribution P=A ∫ vx vy exp-(vx2 + vy2 + vz2) dvx dvy dvz is not zero. I think everybody should realize how bogus some of the authors are. Google...

I'm looking for good examples of physical motivation for integrals over scalar field. Here is an example I've found: If you want to know the final temperature of an object that travels through a medium described with a temperature field then you'll need a line integral It appears to me that...

Homework Statement 1.) Prove that the infinitesimal volume element d3x is a scalar 2.) Let Aijk be a totally antisymmetric tensor. Prove that it transforms as a scalar. Homework EquationsThe Attempt at a Solution [/B] 1.) Rkh = ∂x'h/∂xk By coordinate transformation, x'h = Rkh xk dx'h =...

I'm working with the signature ##(+,-,-,-)## and with a Minkowski space-stime Lagrangian ## \mathcal{L}_M = \Psi^\dagger\left(i\partial_0 + \frac{\nabla^2}{2m}\right)\Psi ## The Minkowski action is ## S_M = \int dt d^3x \mathcal{L}_M ## I should obtain the Euclidean action by Wick rotation. My...

Consider the potential ##U(\phi) = \frac{\lambda}{8}(\phi^{2}-a^{2})^{2}-\frac{\epsilon}{2a}(\phi - a)##, where ##\phi## is a scalar field and the mass dimensions of the couplings are: ##[\lambda]=0##, ##[a]=1##, and ##[\epsilon]=4##. Expanding the field ##\phi## about the point...

Does anyone know any good references for discussion of ##\overline{MS}## theory in phi^4 theory?

Homework Statement Let A be an nxn matrix, and let |v>, |w> ∈ℂ. Prove that (A|v>)*|w> = |v>*(A†|w>) † = hermitian conjugate Homework EquationsThe Attempt at a Solution Struggling to start this one. I'm sure this one is likely relatively quick and painless, but I need to identify the trick...

Homework Statement A self-interacting real scalar field ##\psi(x)## is described by the Lagrangian density ##\mathcal L = \mathcal L_o + \mathcal L_I = \frac{1}{2} (\partial_{\mu}\psi)(\partial^{\mu}\psi) − \frac{1}{2}m^2\psi^2 − \frac{g}{4!}\psi^4 ## where g is a real coupling constant, and...

Homework Statement Consider the following in cylindrical coordinates \rho,\varphi,z. An electric current flows in an infinitely long straight cylindrical wire with the radius R. The magnetic field \mathbf{B} outside of the thread is...

On page 60 of srednicki (72 for online version) for the $$\phi^{3}$$ interaction for scalar fields he defines $$Z_{1}(J) \propto exp\left[\frac{i}{6}Z_{g}g\int d^{4}x(\frac{1}{i}\frac{\delta}{\delta J})^{3}\right]Z_0(J)$$ Where does this come from? I.e for the quartic interaction does this...

The conserved 4-momentum operator for the complex scalar field ##\psi = \frac{1}{\sqrt{2}}(\psi_1 + i\psi_2)## is given in terms of the mode operators in ##\psi## and ##\psi^{\dagger}## as $$P^{\nu} = \int \frac{d^3 p}{(2\pi)^3 }\frac{1}{2 \omega(p)} p^{\nu} (a^{\dagger}(p) a(p) +...

Hi, the following is taken from Peskin and Schroeder page 36: ##\partial_{\mu}\phi(x) \rightarrow \partial_{\mu}(\phi(\Lambda^{-1}x)) = (\Lambda^{-1})^{\nu}_{\mu}(\partial_{\nu}\phi)(\Lambda^{-1}x)## It describes the transformation law for a scalar field ##\phi(x)## for an active...

I'm not really sure if this is true, which is why I want your opinion. I have been trying to prove it, but it will help me a lot if someone can confirm this. Let ## v_{1}, v_{2} ... v_{n} ## be vectors in a complex inner product space ##V##. Suppose that ## | v_{1} + v_{2} +...+ v_{n}| =...

Hello forum. The electric field generated by a changing magnetic field is not conservative. A conservative field is a field with the following features: the closed line integral is zero the line integral from point A to point B is the same no matter the path followed to go from A to B it is...

Does anyone know of an example, preferably a simple one, that can be used to demonstrate that we can have a curvature singularity without a singularity in the Kretschmann scalar?

Homework Statement The aim is to find a solution for the scale factor in a Robertson Walker Metric with a scalar field and a Lagrange multiplier. Homework Equations I have this action S=-\frac{1}{2}\int...

I have a more philosophical question about the interpretation of a mathematical process. We have a chiral superscalarfield shown as partiell Grassmann Integral and transform it into a lagrange. where S and P are real components of a complex scalarfield and D and G are real componentfields of...

why do scalar interactions(for example the higgs vev or its components) reverse the chirality of the interacting particle?? i think this is the key for understanding the mass generation of fermions, but i can't think of a logical reason of the reversed chirality.

I have a question in my textbook where I'm given weight of a "penguin in a sled" as 80N but the object is on a 40 degree angle. Is it telling me that on a normal flat surface the weight is 80N so that way to figure out the force of gravity on the x-axis I must divide the W by 9.8 then plug in my...

Why work done by a force was taken as dot product between force applied and displacement caused?

If R is Ricci scalar ∇i∇j F(R) = ? , where ∇i is covariant derivative.

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...

Homework Statement Homework Equations Mo=Fd Mo=r x F The Attempt at a Solution Alright guys, I did the whole process but I'm pretty sure I just made a little bump somewhere in my calculations which screwed up my answers. First I found everything I could find OA = 350j, so the unit vector...

hi! i need to solve this integral: \rho_s=\int (m/\omega)e^{-\omega/T}d{\vec k} where \omega=\sqrt{m^2+{\vec k}} is the dispersion relation, T is the temperature of the system and m the mass of a particle Thank you!

This question is motivated by one on stack exchange, and on this paper (which comes across a bit student-y but it claims to have been reviewed, and in any case I have reproduced its results in ctensor and gnuplot). So: the KS (abbreviation!) conveys an overview of curvature at a given point in...

I am confused at why ##V_{i,j}V_{j,k}A_{km,i}## the result will end up being a vector (V is a vector and A is a tensor) What are some general rules when you are multiplying a scalar, vector and tensor?

Homework Statement Refer to solution II , the author used the scalar analysis( dot product) to get the direction of moment ...IMO , this is incorrect ... Only cross product can be determined this way . correct me if I'm wrong . Homework EquationsThe Attempt at a Solution

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...

The determinant of a 3x3 matrix can be interpreted as the volume of a parallellepiped made up by the column vectors (well, could also be the row vectors but here I am using the columns), which is also the scalar triple product. I want to show that: ##det A \overset{!}{=} a_1 \cdot (a_2 \times...

Just a question : do we have in Dirac notation $$\langle u|A|u\rangle\langle u|B|u\rangle=\langle u|\langle u|A\otimes B|u\rangle |u\rangle$$ ?

I have a question about the directional derivative of the Ricci scalar along a Killing Vector Field. What conditions are necessary on the connection such that K^\alpha \nabla_\alpha R=0. Is the Levi-Civita connection necessary? I'm not sure about it but I believe since the Lie derivative is...

Homework Statement I'm supposed to prove det A = \frac{1}{6} \epsilon_{ijk} \epsilon_{pqr} A_{ip} A_{jq} A_{kr} using the triple scalar product. Homework Equations \frac{1}{6} \epsilon_{ijk} \epsilon_{pqr} A_{ip} A_{jq} A_{ kr} (\vec u \times \vec v) \cdot \vec w = u_i v_j w_k...

Hello, I am attempting to calculate unit normal and tangent vectors for a scalar field I have, Φ(x,y). For my unit normal, I simply used: \hat{n}=\frac{\nabla \phi}{|\nabla \phi|} However, I'm struggling with using this approach to calculate the unit tangent. I need to express it in terms of the...

Hi, if we adopt the convention, a^{\dagger}_\textbf{p} |0\rangle = |\textbf{p}\rangle then we get a normalization that is not Lorentz invariant, i.e. \langle \textbf{p} | \textbf{q} \rangle = (2\pi)^3 \delta^{(3)}(\textbf{p} - \textbf{q}) . How do I explicitly show that this delta...

Hello, This problem is in reference to the QFT lecture notes (p.18-19) by Timo Weigand (Heidelberg University). He writes: For the real scalar fields, we get self-adjoint operators \phi(\textbf{x}) = \phi^{\dagger}(\textbf{x}) with the commutation relations [\phi(\textbf{x})...

Hello, new to this website, but one question that's been killing me is how can curl of a gradient of a scalar field be null vector when mixed partial derivatives are not always equal?? consider Φ(x,y,z) a scalar function consider the determinant [(i,j,k),(∂/∂x,∂/∂y,∂/∂z),(∂Φ/∂x, ∂Φ/∂y, ∂Φ/∂z)]...

Puting a minus in front of the momentum in the field expansion gives ##\phi \left( {\bf{x}} \right) = \int {{d^3}\tilde p} \left( {{a_{\bf{p}}}{e^{i{\bf{p}} \cdot {\bf{x}}}} + a_{\bf{p}}^ + {e^{ - i{\bf{p}} \cdot {\bf{x}}}}} \right){\rm{ }}\phi \left( {\bf{x}} \right) = \int {{d^3}\tilde...

Hello, I'm reading Tong's lecture notes on QFT and I'm stuck on the following problem, found on p.11-12. A scalar field \phi , under a Lorentz transformation, x \to \Lambda x , transforms as \phi(x) \to \phi'(x) = \phi(\Lambda^{-1} x) and the derivative of the scalar field transforms...

From a textbook. proof that the scalar product ##A\centerdot B## is a scalar: Vectors A' and B' are formed by rotating vectors A and B: $$A'_i=\sum_j \lambda_{ij} A_j,\; B'_i=\sum_j \lambda_{ij} B_j$$ $$A' \centerdot B'=\sum_i A'_i B'_i =\sum_i \left( \sum_j \lambda_{ij} A_j \right)\left( \sum_k...

When one considers a Lorentz transformation between two frames ##S## and ##S'##, such that the coordinates in ##S## are given by ##x^{\mu}## and the coordinates in ##S'## are given by ##x'^{\mu}##, with the two related by x'^{\mu}=\Lambda^{\mu}_{\;\;\nu}x^{\nu} then a scalar field ##\phi (x)##...

Two lines A and B. The angle between them is θ, their direction cosines are (α,β,γ) and (α',β',γ'). Prove, ON GEOMETRIC CONSIDERATIONS: ##\cos\theta=\cos\alpha\cos\alpha'+\cos\beta\cos\beta'+\cos\gamma\cos\gamma'## I posted this question long ago and i was told that this is the scalar product...

I've started f(T) theory but I have a simple question like something that i couldn't see straightforwardly. In Teleparallel theories one has the torsion scalar: And if you take the product you should obtain But there seems to be the terms like . How does this one vanish? because we know...

What is the scalar product of orthonormal basis? is it equal to 1 why is a.b=ηαβaαbβ having dissimilar value

Homework Statement How does the scalar product of displacement four vector with itself give the square of the distance between them? Homework Equations (Δs)2= Δx.Δx ( s∈ distance, x∈ displacement four vector) or how ds2=ηαβdxαdxβ The Attempt at a Solution Clearly I am completely new to the...

First of all, I copy the text in my lecture note. - - - - - - - - - - - - - - - - - - - In general, $$e^{-iTH}$$ cannot be written exactly in a useful way in terms of creation and annihilation operators. However, we can do it perturbatively, order by order in the coupling $$ \lambda $$. For...

Hey, all. I have a question concerning the treatment and use of vectors when solving problems (or in general, really). I know that vectors have both magnitude and direction, while scalars only have magnitude. However, in solving problems and looking at how others have solved them, I've noticed...

we know work is F.s which is a scalar quantity. but why it is defined this way? work actually has something to do with direction. Negative work means reduction in velocity and velocity has direction. What was the problem of defining work as vector that lies in the direction of displacement??

Question is regarding Scalars and Vectors Article. Q: One of the two forces is double the other and their resultant is equal to the greater force . The angle between them is ? Ans : Its answer is cos^-1 (-1/4) My solution : The formula for Cosine law is R= √A²+B² +2ABcos theta Our teacher...

Is it possible to have a free massless scalar field in 1+1 spacetime and then add another field of the right type which interacts with some adjustable strength with the massless field to give mass to the massless field? Is there a Higgs-like mechanism in 1+1 spacetime? Thanks for any help!