Scalar Definition and 829 Threads

Hello, I am having trouble deriving the equation of motion for the quintessence field. The equation of motion which I am meant to get at the end point is: (with ' denoting derivative w.r.t time) φ'' + 3Hφ' + dV/dφ = 0 Using the inflaton lagrangian: (although with a generic potential V(φ)...

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'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...

Hi, I have this math problem where I need to find the scalar component of acceleration at a given time under certain conditions. Usually these problems aren't bad for me, but this one has left me scratching my head. Its giving me ||a|| = 4 and (aT)(T) = 5i +5j -k I have the formula aN =...

In Peskin and Schroeder problem 10.1 is about showing that superficially divergent diagrams that would destroy gauge invariance converge or vanish. We are supposed to prove it for the 1-photon, 3-photon, and 4-photon vertex diagrams. Does this change for scalar QED?

Meant as element of Hilbert space of L^2 functions... etc., the wave function is a vector. In the abstract description with kets and operators on these, the wave function is the scalar product between a ket |Psi> and the "eigenkets" |x> of the position operator: psi(x) = <x|Psi>. So: psi is a...

The magnetic field has no net source or sinks i.e. number of sources are equal to the number of sinks. Can a scalar field also have no net source? Or a source is required for a scalar field?

Can a scalar field exert a torque on a particle?

I was curious, if you were given a vector field F(x,y,z) = <Fx(x,y,z), Fy(x,y,z), Fz(x,y,z)>, and then some scalar function f(x,y,z), how would you define a function θ(x,y,z) of the angle θ between the scalar function and the vector field at any given point. I know how I would find this at a...

Homework Statement Page 16 (attached file) \frac{dH}{dt}|_{t=0} = Δ_{Σ}φ + Ric (ν,ν)φ+|A|^{2}φ \frac{d}{dt}(dσ_{t})|_{t=0} = - φHdσ H = mean curvature of surface Σ A = the second fundamental of Σ ν = the unit normal vector field along Σ φ = the scalar field on three manifold M φ∈C^{∞}(Σ)...

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...

Consider the following integral that comes out of a loop calculation along with some fermionic propagators (e.g virtual one loop correction to a ##p \gamma^* \rightarrow p'## process such as in DIS): $$ \int \frac{\text{d}^d l}{l^2 (l-p)^2 (p+q-l)^2} \text{Tr}(\not p \gamma^{\nu} (\not p + \not...

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}| =...

Reading Geroch's "What is a Singularity in General Relativity?", it seems that polynomial scalar invariants constructed from the Riemann tensor can diverge if we are at infinite distance, and not in a true singularity. Can someone give an example of space-time whose scalar invariant diverges...

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...

Homework Statement Compute $$T_{\mu\nu} T^{\mu\nu} - \frac{T^2}{4}$$ For a massless scalar field and then specify the computation to a spherically symmetric static metric $$ds^2=-f(r)dt^2 + f^{-1}(r)dr^2 + r^2 d\Omega^2$$Homework Equations $$4R_{\mu\nu} R^{\mu\nu} - R^2 = 16\pi^2 \left(...

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})...