Scalar Definition and 829 Threads

If we take the Lagrangian of a spin-0 scalar field and use the Euler-Lagrange equation, we end up with the Klein-Gordon equation. Does that mean that the wave equation of spin-0 scalar particles is the Klein-Gordon equation?Thank you

Let's say I have two vector fields a(x,y,z) and b(x,y,z). Let's say I have a scalar field f equal to a•b. I want to find a clean-looking, simple way to express the directional derivative of this dot product along a, considering only changes in b. Ideally, I would like to be able to express...

Why is ##\theta## a scalar whereas ##d\theta## a vector ?

Hi everyone, I've been looking at a problem that seems simple at first, but appears to be deceptively difficult (unless I'm missing something). 1. Homework Statement I've been looking at a problem that involves the diffusion of a scalar quantity, ##q(x)##, on the semi-infinite domain, ##\leq...

In string theory, if we have NN BCs along ##X^i, i = 1, \ldots, n-1## and DD BCs along ##X^a, a = n, \ldots, 25## then you get, from ##\alpha^{i,a}_{-1}|0,p\rangle ##, ##n## massless vectors and ##24-n## massless scalars. I understand that for the first excited level, ##M^2=0## and so we have...

I am a little bit confused on Center of Mass and Center of Gravity about what are they vectors or scalars . As I may think they are neither because they are simply two points . Am I saying right or wrong .

I was taught a scalar is a quantity that consists of a number (positive or negative) and it might include a measuring unit, e.g. 6, 5 kg, -900 J, etc. I was wondering if complex numbers like 3 + 7j (where j is the square root of minus 1) can be considered scalar quantities too, or is it that...

Consider the Lagrangian $$\mathcal{L}=\frac{1}{2}\partial_{\mu}h\partial^{\mu}h-\frac{1}{2}m^{2}h^{2}-\frac{\lambda}{3!}h^{3}$$ for a real scalar field ##h##. This is the Klein-Gordon Lagrangian with a cubic self-interaction term. Does this model allow the decay process $$h \rightarrow h +...

Homework Statement Suppose we have a covariant derivative of covariant derivative of a scalar field. My lecturer said that it should be equal to zero. but I seem to not get it Homework Equations Suppose we have $$X^{AB} = \nabla^A \phi \nabla^B \phi - \frac{1}{2} g^{AB} \nabla_C \phi \nabla^C...

When I induce magnetic flux through a closed loop, I should expect the lines of flux produced by current in that loop to oppose the change of flux through that loop. But what happens when that loop, say a rectangular loop, is curved into the shape of the letter J (like a candy cane) and my flux...

The theory of a complex scalar field ##\chi## is given by $$\mathcal{L}=\partial_{\mu}\chi^{*}\partial^{\mu}\chi-m_{\chi}^{2}\chi^{*}\chi.$$ Why is it not common to include a factor of ##\frac{1}{2}## in front of the complex ##\chi## kinetic term? What is the effect on the propagator of...

## L(x) = L(\phi(x), \partial_{u} \phi (x) ) = -1/2 (m^{2} \phi ^{2}(x) + \partial_{u} \phi(x) \partial^{u} \phi (x))## , the Lagrange density for a real scalar field in 4-d, ##u=0,1,2,3 = t,x,y,z##, below ##i = 1,2,3 =x,y,z## In order to compute the Hamiltonian I first of all need to compute...

## L(x) = L(\phi(x), \partial_{u} \phi (x) ) = -1/2 (m^{2} \phi ^{2}(x) + \partial_{u} \phi(x) \partial^{u} \phi (x))## , the Lagrange density. ## S= \int d^{4}(x) L (x) ##, the action. ## \phi -> \phi + \delta \phi ## (just shortened the notation and dropped the x dependence) I have ##...

Consider $X, Y$ as $n \times n$ matrices, I am given this definition of scalar product: $$\langle X, Y \rangle = tr(X Y^T),$$ and I need to prove that it is positive definite scalar product. Of several properties I need to prove, two of them are (1) $\langle X, X\rangle \geq 0$ and (2)...

Is angle a vector or a scalar quantity? (eg- θ in l=rθ)

I know that a vector quantity or simply a vector is a physical quantity which has a magnitude and is associated with some definite direction. According to this definition should not pressure and current be vectors since both are associated with some definite direction?In some places they are...

Is it possible to take the "Eistein Scalar" from the Einstein Tensor, like one can take the Ricci Scalar from the Ricci Tensor? If so, is the G of Newton's law the same as this "Einstein Scalar" or is it just the same symbol used in very different things. (Sorry for my bad English. I think...

Hey guys, I got the following derivation for some physical stuff (the derivation itself is just math) http://thesis.library.caltech.edu/5215/12/12appendixD.pdf I understand everything until D.8. After D.7 they get the eigenvalue and eigenvectors from ε. The text says that my δx(t) gets aligned...

Homework Statement Let T(r) be a scalar field. Show that, in spherical coordinates ∇T = (∂T/∂r) rˆ + (1/r)(∂T/∂θ) θˆ + (1/(r*sin(θ)))(∂T/∂φ) φˆ Hint. Compute T(r+dl)−T(r) = T(r+dr, θ+dθ, φ+dφ)−T(r, θ, φ) in two different ways for the infinitesimal displacement dl = dr rˆ + rdθ θˆ + r*sin(θ)dφ...

What exactly are short-distance quantum corrections? Why is the mass term of a fundamental scalar field highly sensitive to short-term quantum corrections?

Consider a quantum scalar field theory with interaction terms of the form ##\phi^{n}##, where ##n## is not an integer. Where are some examples of physical theories which involve such interaction terms?

Homework Statement Prove that the Noether charge ##Q=\frac{i}{2}\int\ d^{3}x\ (\phi^{*}\pi^{*}-\phi\pi)## for a complex scalar field (governed by the Klein-Gordon action) is a constant in time. Homework Equations ##\pi=\dot{\phi}^{*}## The Attempt at a Solution...

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...

I find it difficult to believe that the canonical commutation relations for a complex scalar field are of the form ##[\phi(t,\vec{x}),\pi^{*}(t,\vec{y})]=i\delta^{(3)}(\vec{x}-\vec{y})## ##[\phi^{*}(t,\vec{x}),\pi(t,\vec{y})]=i\delta^{(3)}(\vec{x}-\vec{y})## This seems to imply that the two...

I am well-versed with the definition of scalar and vector quantities.The confusion I mainly have is at many points, my textbook makes ambiguous statements like "Because force is vector quantity it follows that field strength is also a vector quantity." What relationship should arbitrary...

Hey guys. So, as i was going through Griffith's Electrodynamics, and i came across this problem: In the solutions: How to they actually get to that expression for V = (V(bar)+vAx(bar) )Ɣ? I understand everything after that, but this just made me very confused. How do they get this from the...

Homework Statement Construct a subset of the x-y plane R2 that is (a) closed under vector addition and subtraction, but not scalar multiplication. Hint: Starting with u and v, add and subtract for (a). Try cu and cv Homework Equations vector addition, subtraction and multiplication The...

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...

I recently learned that the general formula for the dot product between two vectors A and B is: gμνAμBν Well, I now have a few questions: 1. We know how in Cartesian coordinates, the dot product between a vector and itself (in other words A ⋅ A) is equal to the square of the magnitude |A|2...

Book: Landau Lifshitz, The Classical Theorey of Fields, chapter 11, section 95. I have gone through the derivation of Einstein field equations but not without holes to fill and fix in my understanding. Let's start with the action for the grtavitational field ##S_g## which after some explanation...

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...

Scalar Product is defined as ##\mathbf A \cdot \mathbf B = | \vec A | | \vec B | \cos \theta##. With the construct of a triangle, the Law of Cosines is proved. ##\mathbf A## points to the tail of ##\mathbf B##. Well, ##\mathbf C## starts from the tail of ##\mathbf A## and points to somewhere...

I've been asked by someone with minimal background in physics to explain what vector and scalar quantities are and give examples. Here are my thoughts: A scalar is a quantity that has a magnitude only, it is completely specified by a single number. Importantly, it has no directional dependence...

Homework Statement for the line segment c2 , why did the author want to differentiate dx with respect to dy ? and he gt dx = 0 ? I'm curious why did he didnt do so for C3 , where dy= 0 ..Why didnt he also differentiate dy with dx ? dy/dx = 0 ? Homework EquationsThe Attempt at a Solution is...

Consider a rope wraps an angle ##\theta## around a pole with a coefficient of static friction ##\mu##. You pull one end with a tension ##T_0##. The force that the other end can support is given by ##T=T_0e^{\mu\theta}##(derivation below). For ##\mu=1## and ##\theta=2\pi##, ##T=530T_0##. That...

Homework Statement ## \frac{d}{dt}\gamma(t)\vec{u(t)} ## Homework Equations See above The Attempt at a Solution This comes from trying to verify a claim in Chapter 12 of Griffiths Electrodynamics, 4th. edition (specifically Eq. 12.62 -> Eq. 12.63, if anyone has it on hand). I would have...

Homework Statement Page 35 of Jackson's Electrodynamics (3rd ed), it gives the following equation (basically trying to prove your standard 1/r potential is a solution to Poisson equation): \nabla^2 \Phi_a = \frac{ -1 }{ \epsilon_0 } \int \frac{ a^2 }{( r^2 + a^2)^{5/2} } \rho( \boldsymbol{x'}...

how is flux scalar if the magnetic field and area both are vector quantities?

According to David Tong's notes the real scalar field can't be coupled to the electromagnetic field because it doesn't have any "suitable" conserved currents. What does "suitable" mean? The real field does have conserved currents, why aren't those suitable?

Homework Statement Determine the vector potential due to a short wire running from (−L/2, 0, 0) to (L/2, 0, 0) carrying a current I = I0cos(ωt) (consider only points at distance r >> L from the origin). (You may neglect the effect of the return circuit.) Now determine the corresponding scalar...

Hi, I am looking for a proof that explains why gradient is a vector that points to the greatest increase of a scalar function at a given point p. http://math.stackexchange.com/questions/221968/why-must-the-gradient-vector-always-be-directed-in-an-increasing-direction I understand the proof...

Hello all, I hope you can give me a hand with a QFT homework I'm working on. We are to compute the beta equation of a Non-abelian SU(N) theory with: Complex scalars (massless), bosons, ghosts. My question is referring to the Boson self-energy scalar loop correction. 1. Homework Statement We...

Note: I am not in the course where this problem is being offered; it was simply an interesting linear algebra "thought question" that I found online to which I believe I have found a solution. However, there is one step in my solution that I am unsure about, so thank you to anyone who spares the...

(Scalar)·(Scalar) = Scalar (Scalar)·(Vector) = Scalar (Vector)·(Vector) = Scalar (Scalar)x(Scalar) = Not valid (Scalar)x(Vector) = Vector (Vector)x(Vector) = VectorDid I get them right, if not why? Thanks

Hey guys, I really need help in revising my Axiom 6 for my Linear Algebra course. My professor said, "You need to refine your statement. You want to show rx1 and rx2 are real numbers. You should not state they are real numbers." Here is my work: Proof of Axiom 6: rX is in R2 for X in R2...

I have a simple question about Lewkowycz and Maldacena's paper http://arxiv.org/abs/1304.4926v2'][/PLAIN] http://arxiv.org/abs/1304.4926v2 In section 2, they consider the scalar field in BTZ background ground and require boundary condition of the scalar field, $\phi \sim e^{i\tau}$ . This...

I have a few questions regarding the derivation of the degree of divergence for feynman diagrams. The result is $$D = [g_E] - \sum_{n=3}^{\infty} V_n [g_n]$$ (following notation in Srednicki, ##P118##) I am trying to understand what ##[g_E]## is here? Since in this set up we are summing over...

Why A.A = ||A||^2 , I know that from product rule we can prove this where theta =0 , I am asking this because I have seen many proves for A.B = ||A||||B||cos(theta) and to prove this they have used A.A = ||A||^2, how can they use this , this is the result of dot product formula. I havee seen...

why do we take cross product of A X B as a line normal to the plane which contains A and B. I also need a proof of A.B = |A||B|cos(theta), I have seen many proves but they have used inter product ,A.A = |A|^2, which is a result of dot product with angle = 0, we can't use this too prove...

Homework Statement I must find the following equation of motion: φ'' + 3Hφ' + dV/dφ = 0 Using the scalar field Lagrangian: (replace the -1/2m^2φ^2 term with a generic V(φ) term though) with the Euler-Lagrange Equation I know that I must assume φ = φ(t) and the scale factor a = a(t)...