This page is about helicity in fluid dynamics. For helicity of magnetic fields, see magnetic helicity. For helicity in particle physics, see helicity (particle physics).In fluid dynamics, helicity is, under appropriate conditions, an invariant of the Euler equations of fluid flow, having a topological interpretation as a measure of linkage and/or knottedness of vortex lines in the flow. This was first proved by Jean-Jacques Moreau in 1961 and Moffatt derived it in 1969 without the knowledge of Moreau's paper. This helicity invariant is an extension of Woltjer's theorem for magnetic helicity.
Let
u
(
x
,
t
)
{\displaystyle \mathbf {u} (x,t)}
be the velocity field and
∇
×
u
{\displaystyle \nabla \times \mathbf {u} }
the corresponding vorticity field. Under the following three conditions, the vortex lines are transported with (or 'frozen in') the flow: (i) the fluid is inviscid; (ii) either the flow is incompressible (
∇
⋅
u
=
0
{\displaystyle \nabla \cdot \mathbf {u} =0}
), or it is compressible with a barotropic relation
p
=
p
(
ρ
)
{\displaystyle p=p(\rho )}
between pressure
p
{\displaystyle p}
and density
ρ
{\displaystyle \rho }
; and (iii) any body forces acting on the fluid are conservative. Under these conditions, any closed surface
S
{\displaystyle S}
on which
n
⋅
(
∇
×
u
)
=
0
{\displaystyle n\cdot (\nabla \times \mathbf {u} )=0}
is, like vorticity, transported with the flow.
Let
V
{\displaystyle V}
be the volume inside such a surface. Then the helicity in
H
{\displaystyle H}
is defined by
H
=
∫
V
u
⋅
(
∇
×
u
)
d
V
.
{\displaystyle H=\int _{V}\mathbf {u} \cdot \left(\nabla \times \mathbf {u} \right)\,dV\;.}
For a localised vorticity distribution in an unbounded fluid,
V
{\displaystyle V}
can be taken to be the whole space, and
H
{\displaystyle H}
is then the total helicity of the flow.
H
{\displaystyle H}
is invariant precisely because the vortex lines are frozen in the flow and their linkage and/or knottedness is therefore conserved, as recognized by Lord Kelvin (1868). Helicity is a pseudo-scalar quantity: it changes sign under change from a right-handed to a left-handed frame of reference; it can be considered as a measure of the handedness (or chirality) of the flow. Helicity is one of the four known integral invariants of the Euler equations; the other three are energy, momentum and angular momentum.
For two linked unknotted vortex tubes having circulations
κ
1
{\displaystyle \kappa _{1}}
and
κ
2
{\displaystyle \kappa _{2}}
, and no internal twist, the helicity is given by
H
=
±
2
n
κ
1
κ
2
{\displaystyle H=\pm 2n\kappa _{1}\kappa _{2}}
, where
n
{\displaystyle n}
is the Gauss linking number of the two tubes, and the plus or minus is chosen according as the linkage is right- or left-handed.
For a single knotted vortex tube with circulation
κ
{\displaystyle \kappa }
, then, as shown by Moffatt & Ricca (1992), the helicity is given by
H
=
κ
2
(
W
r
+
T
w
)
{\displaystyle H=\kappa ^{2}(Wr+Tw)}
, where
W
r
{\displaystyle Wr}
and
T
w
{\displaystyle Tw}
are the writhe and twist of the tube; the sum
W
r
+
T
w
{\displaystyle Wr+Tw}
is known to be invariant under continuous deformation of the tube.
The invariance of helicity provides an essential cornerstone of the subject topological fluid dynamics and magnetohydrodynamics, which is concerned with global properties of flows and their topological characteristics.
In the context of non relativistic quantum mechanics, or better, if I consider the neutrino's mass to be zero, the phrase
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Hello everybody!
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Supposing of being in ultrarelativistic regime, so helicity and...
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How do I relate...
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Homework Statement
For massless particles, we can take as reference the vector ##p^{\mu}_R=(1,0,0,1)## and note that any vector ##p## can be written as ##p^{\mu}=L(p)^{\mu}_{\nu}p^{\nu}_R##, where ##L(p)## is the Lorentz transform of the form
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Hi,
My question basically comes from this book called "Deep inelastic scattering"
In the second chapter, it first made a similar argument for J = 1 Jz = -1, +1 which is pretty easy to get along with. However, immediately following from that there was this argument which confuses me a bit...
The helicity in non relativistic quantum mechanics is given by ##\sigma \cdot p / |p|## where ##\sigma## are the pauli matrices and ##p## the momentum. In spinor space, the ##\sigma## are 2x2 matrices, and thus, the helicity, if we calculate it, is a 2x2 quantity. But in 3d physical space, the...
Hello! I have some questions about helicity and chirality: So I understand how is helicity defined and that it has eigenvalues of 1 or -1. But can a particle (massless) have mixed helicity? Like the spin not to be along the direction of motion? (I assume it can but I want to make sure, because...
In trying to understand the Neutrino where it has mass and its chirality is the same as its helicity, I have always had trouble visualizing a particle. I recently ran into this particle. I believe the "the chirality is the same as helicity" as in one direction it would feed things through the...
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Homework Statement
Let ##V^{3}(t)## be a compact region moving with the fluid.
Assume that at ##t=0## the vorticity ##2##-form ##\omega^{2}## vanishes when restricted to the boundary ##\partial V^{3}(0)##; that is, ##i^{*}\omega^{2}=0##, where ##i## is the inclusion of ##\partial V## in...
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The drive shaft of a truck undergoing acceleration twists a small amount. That twist defines a helicity. If you just know the rotation of a drive shaft you can not tell which way power flows but if given rotation and the twist of the shaft that can determine the direction of power flow?
Thanks!
Why helicity of phon is 1 but not 3 or higher?Is there any quantity relation between the circular polarization of light and spin of photon?Why spin of graviton is 2?Is there any relation with vector and tensor charater of electromagnetic and gravitation fields and of P symmetry?Why do the...
My lecture notes give an example of two decay modes of ##K^+##, namely ##K^+\rightarrow \mu^+ \nu_\mu## and ##K^+\rightarrow e^+ \nu_e##. Both of these decays are suppressed due to helicity considerations which I understand, and the suppression factors are ##\frac{m_\mu c^2}{E_\mu}## and...
Hi there,
The question about the helicity operator ## h= S . \bf{p} ## ( where S is 2 by 2 matrix, with ##\sigma^i ## on the diagonal ), that as mentioned in a reference as [arXiv:1006.1718], it commutes with the Dirac Hamiltonian ## H = \gamma^0 ( \gamma^i p^i + m ) ## equ. (3.3), due to...
Could someone please help me to understand the difference between the concepts of Weak Isospin, Chirality and Helicity. In particular, I have the following questions to which I was unable to find answers so far:
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Homework Statement
Draw feynman diagrams for pi+ muon lepton decay and suggest which process is more likely.Homework EquationsThe Attempt at a Solution
[/B]
The feynman diagrams are:
The lepton decays proceed via the weak interaction W+ boson. This implies that e+ should be...
A left handed neutrino (chirality) can be seen with a right helicity due to a lorentz boost. Can this neutrino interact ? Yes because it is still left-handed chirality ?
(Chiral Representation ##\gamma_5## is diagonal)
According to An introduction to QFT - Peskin & Schroeder 3.3 : ##h=\hat{p}\cdot S##
and ##h=+1/2## is right-handed while ##h=-1/2## is left-handed.
It is quite easy for fermions. But I'm confused when it comes to anti-fermions.
In Pestkin's...
First: a question about spin. When we say that an electron has spin 1/2, we mean that it can have the values ħ/2 or -ħ/2. So when we say that a photon has spin 1, I would expect this to mean that the measurement of a photon would give values either ħ or -ħ. But then I am confused by the...
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What about Sx and Sy? They are both ZERO?
Hi all!
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Good morning everyone ! I've been reading discussions on PF for a long time, but here I'm stuck on a little problem that really annoys me and I couldn't find answer anywhere, so I guess it was time to register. :>
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At the beginning of cpt 9, Griffiths states that massive bosons have three polarization states (m_s = 1, 0, -1), but massless ones have only two (m_s = 1, -1). Are these polarization states the same thing as helicity states? I.e. the W/Z would have 3 helicity states and the photon only 2?
1) Since Wigner it is well known that for massless particles of spin s the physical states are labelled by helicity h = ±s; other states are absent. So e.g. for photons the physical states are labelled by |kμ, h> with kμkμ = 0 and h = ±1 and we have two d.o.f.
2) For gauge theories with...
Dear everyone,
I have a simple question about the helicity of photon. The helicity operator is defined as
\hat{\mathbf{S}} \hat{\mathbf{p}}/|\mathbf{p}|. How to show the photon has +1/-1 helicity eigenvalue from this definition?
Thank you ~~
Hi...
I read somewhere that positron, in the massless, limit will have the same helicity as the antineutrino.
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So helicity operator must commute with the SU(2) generators.
Please confirm.
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In page 72, equation (2.5.39) gives
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and he says \sigma will be the helicity. As he explains:
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Hi, I have a problem involving helicity.
Homework Statement
In a scattering
e^{+}+e^{-} \rightarrow \nu_{\mu}+\overline{\nu}_{\mu}
I have to determine for which values of the helicity of initial particles the cross section is not 0.
Homework Equations
The Attempt at a Solution
On a...
Hello all,
This is something that has irked me for a while. The Li/Yang/Wu beta decay showed parity violation in the weak force, but from what I gather, it was the helicities of the electrons they measured, while it is the chiral states which are important. For a massive fermion, aren't the...
Can somenone check if my reasoning is correct. I would like to have deeper
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Let's consider two particles:
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(these informations...
Hi.
I need some information about chirality and Helicity
basic information
it's application in Dirac equation and quantum field
any notes or presentation
good reference
I'm still Beginner in these subject
Hi, I was reading a lecture of qft and I found that two equations:
\begin{flalign*}
i \gamma^\mu \partial_\mu\psi_R - m\psi_L=0 \\
i \gamma^\mu \partial_\mu\psi_L - m\psi_R=0
\end{flalign*}
after splitting in two Dirac's equation with Weyl's projectors.
I found that really interesting that the...
Hello,
1)Why does the number of helicity states depend on space-time dimension ?
Masselss graviton in 4 dimensional space has 2 helicity states ( 2 degrees of freedom). In 5 dimensional space it has (still massless in 5 D) 5 helicity states (5 degrees of freedom) ...
In 6 dimesional...
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I've had a look at this...
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Yet, it seems to me that references keep talking of Weyl spinors as if they have well-defined helicities, regardless of the mass...
All neutrinos are left handed and all anti neutrinos are right handed.so,it should be lorentz invariant and travel at speed of light. if it travels at c,then it is massless. but, neutrino oscillation requires mass? why there is contradiction?
Hi...
Consider a neutrino with a Dirac mass m_\nu and the weak interaction
{\cal{L}}=\frac{g}{2 \sqrt{2}} \sum_l[{W_{\mu}^+ \cdot \bar{\psi}_{\nu_l} \gamma^{\mu}(1-\gamma_5)\psi_l + W_{\mu}^- \cdot \bar{\psi}_{l} \gamma^{\mu}(1-\gamma_5)\psi_{\nu_l} }\right{]} + \frac{g}{4...
I'm perplexed about something that Wikipedia says about photon helicity:
(see http://en.wikipedia.org/wiki/Photon)
But for a photon, doesn't the spin vector always point in the same direction as the momentum vector - and therefore, shouldn't the magnitude of a photon's helicity equal it's...