What is Scalar field: Definition and 206 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.

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  1. M

    Scalar Field Theory-Vacuum Expectation Value

    Homework Statement I am given an equation for a quantized, neutral scalar field expanded in creation and destruction operators, and need to find the vacuum expectation value of a defined average field operator, squared. See attached pdf. Homework Equations Everything is attached, but I...
  2. R

    Complex scalar field - Feynman integral

    Homework Statement For a real scalar field \phi, the propagator is \frac{i}{(k^2-m_\phi^2)}. If we instead assume a complex scalar field, \phi = \frac{1}{\sqrt{2}} (\phi_1 + i \phi_2), where \phi_1,\phi_2 are real fields with masses m_{\phi 1},m_{\phi 2}, what is the propagator...
  3. O

    Scalar field as quantum operator.

    Hallo, I was wondering what is the physical significance of scalar field \Phi (x) as an quantum operator. \Phi (x) have canonical commutation relation such as [ \Phi (x) , \pi (x) ] so it must be an opertor, thus what are his eigenstates? Thanks, Omri
  4. P

    Computing Thermal Average for a Free Real Scalar Field

    Homework Statement Homework EquationsHi! I read a book where a free real scalar field with Hamiltonian H = \int \dfrac{\mathrm{d}^{3}p}{(2 \pi)^{3}} \, E_{\vec{p}} a_{\vec{p}}^{\dagger} a_{\vec{p}} is beeing discussed. Note that: E_{\vec{p}} = \sqrt{\vert \vec{p} \vert^{2} + m^{2}}...
  5. M

    Derive Feynman Rules for Complex Scalar Field

    Homework Statement Derive the Feynman rules for for a complex scalar field. Homework Equations L=\partial_\mu\phi^\dagger\partial^\mu\phi +m^2\phi-\lambda/4 |\phi|^4The Attempt at a Solution I wrote the generating functional for the non-interacting theory Z_0[J]=Z_0[0]exp(-\int...
  6. X

    Massive Scalar Field in 2+1 Dimensions

    Homework Statement We wish to find, in 2+1 dimensions, the analogue of E = - \frac{1}{4\pi r} e^{-mr} found in 3+1 dimensions. Here r is the spatial distance between two stationary disturbances in the field. Homework Equations In 3+1 we start from E = - \int \frac{ d^3 k }{(2\pi)^3}...
  7. jfy4

    Complex Scalar Field and Probability Field

    Hi, I was looking at the lagrangian and conserved currents for the free complex scalar field and it looks like it has a striking similarity to the conserved current for probability: \frac{\partial \rho}{\partial t}=\nabla\cdot \vec{j} where j_i =-i(\psi^{\ast}\partial_i \psi -...
  8. W

    Complex scalar field and contraction

    Hi guys, If I use the definition of the scalar complex field as the combination of two scalar real fields, I can get \phi (x) = \int \frac{d^3 p}{(2\pi )^3} \frac{1}{\sqrt{2p_0}} [ \hat a _{\vec{p}} e^{-ip.x} + \hat b _{\vec{p}}^{\dagger } e^{ip.x}] which I can rewrite in terms of...
  9. R

    Spectral weight function and the mass shift of a scalar field

    In the Kallen-Lehmann spectral representation (http://en.wikipedia.org/wiki/K%C3%A4ll%C3%A9n%E2%80%93Lehmann_spectral_representation) the interacting propagator is given as a weighted sum over free propagators. The pole of the integracting propagator is, of course, given by p^2=m^2, m being the...
  10. 7

    Exact differential of scalar field

    Suppose I have the scalar field f in the xy-plane and that it is smooth. Its total derivative is given the normal way, i.e. df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy and the gradient of f is given the normal way as well. I read in a paper that, due to the...
  11. A

    Is This Calculation of Grad \(\psi\) for \(\psi(x,y,z) = (y-1)z^2\) Correct?

    For the following scalar field: \psi(x,y,z) = (y-1)z2 Find grad \psi Here is my attempt at: Multiplying out brackets: yz2 - z2 Therefore grad \psi = 0+Z2 J -2ZK Is this correct??
  12. O

    What is the difference between scalar and vector functions?

    Hi all :) can anybody help me out in understanding scalar function and vector function? the difference between them
  13. L

    Calculating the Effective Action of a Scalar Field Theory

    The effective action Γ[ϕ] for a scalar field theory is a functional of an auxiliary field ϕ(x). Both Γ and ϕ are defined in terms of the generating functional for connected graphs W[J] as W[J] + \Gamma[\phi] = \int d^dx J \phi , \quad \frac{\delta}{\delta J(x)} W[J] = \phi(x) Show - \int...
  14. C

    Differentiating the complex scalar field

    Basic question on scalar filed theory that is getting on my nerves. Say that we have the langrangian density of the free scalar (not hermitian i.e. "complex") field L=-1/2 (\partial_{\mu} \phi \partial^{\mu} \phi^* + m^2 \phi \phi^*) Thus the equations of motion are (\partial_{\mu}...
  15. P

    Understanding Gradient Vector of Scalar Field (grad)

    Dear All I am having trouble understanding the gradient vector of a scalar field (grad). I understand that you can have a 2D/3D space with each point within that space having a scalar value, determined by a scalar function, creating a scalar field. The grad vector is supposed to point in...
  16. C

    BRS: Static Axisymmetric Gravitationless Massless Scalar Field Solutions

    BRS: Static Axisymmetric "Gravitationless" Massless Scalar Field Solutions This thread is an (easy and amusing) companion to a previous BRS, "The Weyl Vacuums" www.physicsforums.com/showthread.php?t=378662 I. The Family of "Gravitationless" Solutions I will describe a family of...
  17. N

    Scalar field theory - Feynman diagrams and anti-particles

    I'm working on a "draw all possible Feynman diagrams up to order 2" problem for a scalar field that obeys the Klein-Gordon equation, and I'm wondering about a few things. When I did a course on particle physics and was first introduced to Feynman diagrams in the context of QED (but not QED...
  18. N

    Scalar field energy for two delta function sources

    I'm trying to evaluate the energy shift in a scalar field described by the Klein-Gordon equation caused by adding two time-independent point sources. In Zee's Quantum Field Theory in a Nutshell, he shows the derivation for a (3+1)-dimensional universe, and I'm trying to do the same for an...
  19. 3

    Interaction scalar field invariance

    The problem: http://www.gobigbang.nl/cft/cft.jpg An attempt of the solution: http://www.gobigbang.nl/cft/DSCN2722.JPG My problem is question c. I don't have a clue how to see that the improved current is conserved... Can anyone help me? Update I managed to solve the question by using the...
  20. V

    Clarifying Boundary Conditions and Scalar Field Quantization in QFT

    This commmunity has so many nice people, so helpful, I am learning QFT from Srednicki I would be glad if some one can clarify, all the books talk about boundary conditions which are finite at spatial infinity and give the general solution for canonical quantization of scalar field, 1) how...
  21. P

    What's the classical picture of phi^4 scalar field theory?

    I know that the classical picture of QED is Coulomb interaction, magnetic interaction etc. But what does the classical phi^4 theory look like? In particular, do particles attract or repel each other in this theory? P.S. I'm surprised that my field theory books never discuss this. (At least in...
  22. M

    Divergence of a vector field is a scalar field?

    Hello. How can I show the Divergence of a vector field is a scalar field(in E^{3}) ? Should I show that Div is invariant under rotation? x^{i'}=a^{ij}x^{j},V^{'}_{i}(\stackrel{\rightarrow}{x})=a_{ij}v_{j}(\stackrel{\rightarrow}{x}) then \frac{\partial...
  23. B

    Interpreting g(f u\otimes u + v\otimes v) with Scalar Field f

    I know that: g(a u\otimes v) = a g(u\otimes v) where u and v are vectors and a is a constant, but what if a is a scalar field, is this rule also true? ie. how do I interpret the expression: g(f u\otimes u + v\otimes v) where u and v are vector fields and f is a scalar field?
  24. R

    Units of Scalar Field \phi & Lagrangian Density

    What are the dimensions of a scalar field \phi ? The Lagrangian density is: \mathcal L= \partial_\mu \phi \partial^\mu \phi - m^2 \phi \phi So in order to make all the terms have the same units, you can try either: \mathcal L=\frac{\hbar^2}{c^2} \partial_\mu \phi \partial^\mu \phi -...
  25. T

    How to Determine Time Ordering in Phi-3 Theory for a 2-to-3 Particle Process?

    i have been given a problem for writing s matrix in second order perturbative theory for an interaction hamiltonian with phi 4 and phi 3 contributions. it is also given that our initial state is of 2 particles and final state is of three particles. now in solving that i have to take time...
  26. M

    Gradient of a scalar field in a given direction

    I have to find the gradient of a scalar field, h, at a certain point in a direction given by a vector. I know, \vec{\nabla}h will give me the direction of maximum slope, and its magnitude is the magnitude of the slope, but i don't know where to start in finding the slope in any other...
  27. B

    Is a scalar field incompatible with the Principle of Equivalence?

    As I know, Einstein initially tried describe the gravitational interaction as mediated by a scalar field, but he later gave up this idea because it is incompatible with the Principle of Equivalence.I don't know how this idea is incompatible with the Principle of Equivalence. Please help me. Thanks
  28. D

    Understanding Scalar Fields and the Laplace Equation: How Do They Relate?

    I've recently read about Null Identities of vector analysis. I'm having a problem in understanding what is it by "taking the curl of the grad of any scalar field is equal to zero." What is by definition of scalar field then? How would it looks like? Is position vector a scalar field? If No...
  29. nicksauce

    Total Momentum Operator for Free Scalar Field

    Homework Statement I want to show that \mathbf{P} = -\int d^{3}x}\pi(x)\nabla\phi(x) = \int{\frac{d^{3}p}{(2\pi)^3}\mathbf{p}a_{p}^{\dagger}a_p for the KG field.Homework Equations \phi(x) = \int{\frac{d^{3}p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_p}}(a_p + a_{-p}^{\dagger})e^{ipx} \pi(x) =...
  30. C

    Sketching the Gradient of a Scalar Field: How to Implement and Interpret?

    Homework Statement Calculate the gradient of the scalar field f(x,y) = x^{2} - y^{2} . Sketch the gradient for a few point on two straight lines y = x and y = -x on the plane and comment on the properties of the sketch. Homework Equations The Attempt at a Solution So I worked...
  31. S

    Weyl invariant scalar field theory

    I'm not sure if this is the right place for this question, so feel free to move it. Anyway, my question is, is there any good reason why the following field theory should be Weyl invariant in an arbitrary dimension d>1: S = \int d^d x \sqrt{g} \left( g^{\mu \nu} \partial_\mu \phi \partial_\nu...
  32. C

    Question About Complex Scalar Field: Advantages/Disadvantages?

    I wanted to ask a quick question about the complex scalar field. My question is that does the scalar field need to be complex in order to include the part for anti-particles or do you regards the scalar field for particles and anti-particles seperate. I saw this specifically when you second...
  33. I

    [QFT] Feynman rules for self-interacting scalar field with source terms

    I'm not sure if this is the right place to post a graduate level course material, but I have a question about perturbative expansion of the 2n-point function of a scalar field theory. Homework Statement First, the question: In which space (position or momentum) is the topological distinctness...
  34. M

    Question about quantization of scalar field

    Why the quantization of scalar field resolves the energy negative problem that exist in the klein-gordon equation?
  35. D

    Why is the Higgs Field a Scalar Field? Exploring Its Nature

    Why is the Higgs field a scalar field? I understand if it is one, it will have no spin and no angular momentum. But understanding that a particle is a scalar seems to me a leap of faith. What am I not getting?
  36. P

    Deriving the Poincare algebra in scalar field theory

    Homework Statement Find the commutators [P^\sigma,J^{\mu \nu}] The answer is part of the Poincare algebra [P^\sigma,J^{\mu \nu}]=i(g^{\mu \sigma}P^\nu-g^{\nu \sigma}P^\mu) If someone can convince me that \partial_i T^{0\mu} = 0, (i.e. the energy-momentum tensor has no explicit spatial...
  37. I

    Variation with scalar field coupled to gravity in 2D

    The two dimensional action is: S_k = \int d^2\sigma\sqrt{h}\left(\partial_\alpha\phi\partial^\alpha\phi - \frac{i}{2}kR^{(2)}\phi\right) where k is a constant, R^{(2)} is the two dimensional scalar curvature. I'm trying to derive the following energy momentum tensor: T_{\alpha\beta}^k =...
  38. K

    Energy-momentum tensor for a scalar field (sign problem)

    Hi I have a small subtle problem with the sign of the energy-momentum tensor for a scalar field as derived by varying the metric (s.b.). I would appreciate very much if somebody could help me on my specific issue. Let me describe the problem in more detail: I conform to the sign convention...
  39. A

    Can photon couple to scalar field?

    I have seen in one paper that photon is coupled to dilaton field which is scalar and motivated by string theory. I do not understand this. Photon is carrier of electromagnetic field and so I thought it can only couple to electrically charged fields. Can anyone explain?
  40. M

    Explicit expressions for creation/annihilation operator of the free scalar field

    I've been trying to work my way through some of my lecture notes, and have hit this snag. (n.b. I use k_0 \equiv +\sqrt{\vec{k}^2 + m^2}) We have a(q) = \int d^3 x e^{iqx} \{ q_0 \phi(x) + i \pi(x) \} a^{\dagger}(q) = \int d^3 x e^{-iqx} \{ q_0 \phi(x) - i \pi(x) \} To calculate the...
  41. K

    Energy momentum tensor of a scalar field by varying the metric

    Suppose you are given the Lagrangian of a scalar field \Phi(t) \mathcal{L} = \frac{1}{2} \dot{\Phi}- \nabla \Phi - V(\Phi ). By introducing covariant notation with \eta_{\mu \nu} = (1,-1,-1,-1) this reads as \mathcal{L} = \frac{1}{2} \eta^{\mu \nu} \partial_\mu\Phi \;\partial_\nu\Phi-...
  42. R

    RS 1, massless scalar field, and sep of vars.

    Hey, this is rather involved but I hope someone can help me out. I am reading http://arxiv.org/abs/0704.3626 ( the casimir force in randall sundrum models) and am trying to get from equation 2.1 : g^{\mu\nu}\partial_\mu\partial_\nu\Phi+e^{2ky}\partial_y(e^{-4ky}\partial_y\Phi)=0 to...
  43. B

    Coupling fermions to a scalar field: Interpretation problem

    Hi all, I have a little problem concerning the coupling of a fermion to CP^N (or better a 2D scalar O(3) model). Its not a mathematical type of problem. I just read on "The coupling of fermions to the three-dimensional noncommutative $CP^{N-1}$ model: minimal and supersymmetric extensions"...
  44. R

    Exploring the Kinetic Term of a Scalar Field

    I've been wondering about terms you typically find in the action of a field theory, for example consider the kinetic term of a scalar field S=\int d^4x(\partial_\mu\phi\partial^\mu\phi). I've read that it can be thought of as the kinetic energy of the field - but this just doesn't sit to...
  45. C

    Consider a surface S on which a scalar field f is defined

    "Consider a surface S on which a scalar field f is defined" "Consider a surface S on which a scalar field f is defined" what does "on which is defined" mean phys descript answers appreciated!
  46. C

    Scalar field and spin 1/2 field

    are bosons represented by a scalar field and fermions represented by a spin 1/2 field or how does it work?
  47. H

    Gauge group SU(5) coupled to a scalar field

    1. For a project on elementary particle physics I have to consider a gauge theory with the gauge group SU(5) coupled to a scalar field. I am to use a certain non-zero vacuum expectation value for the scalar field and check what happens to the gauge bosons. I have already done this for...
  48. J

    How Can I Use Stokes' Theorem to Show Integral of fgrad(g)*dr=0?

    Homework Statement Let S be a simple parametrically defined surface with boundary C as in Stokes' Theorem. Let f and g be two continuously differentiable scalar fields defined on S. Let n be a choice of unit normal on S. If grad(f) is perpendicular to grad(g) x n everywhere on S, show that...
  49. J

    Redefinition of a scalar field

    Hello everyone. I was hoping that someone could possibly help me with a problem I've got. If you have an action for two independent scalar fields, say A and B (arbitrary functions of (x_mu), both without any zeros), then can I redefine the action in terms of two new scalar fields A and C=AB...
  50. S

    Scalar field pressure and energy density

    Hi all, I'm hoping someone can help me out as I'm really stuck. With reference to the top of page 7 at http://faculty.washington.edu/mrdepies/Survey_of_Dark_Energy2.pdf I'd like to know how to get the quoted energy density and pressure of phi from the stress-energy tensor. I am very...
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