The spin of a field

Question

I have searched for an explanation for the math behind the spin of fields, such as the electromagnetic field has a spin of 1 and the gravitational field has a spin of 2.
The internet did not provide a satisfying helpful explanation. I understand that it has to do with repulsion and attraction of different charges but I don't fully understand it. Can someone explain the math to me, please?

Answer 1

I would recommend A. Zee's "QFT in a Nutshell" chapter 1.5 or Peskin & Schroeder pp.125-126. Meanwhile, Zee assumes a small mass for force carriers and this gives rise to the famous vDVZ discontinuity for gravity. Yet, the key lesson remains intact, so I think you can just refer to Zee's textbook. Anyhow, I will discuss massless carriers here in my point of view.

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Basics of Quantum Amplitudes

ㅤㅤAt the fundamental level of our universe, what we conceive as "classical forces" are in fact implemented from "digital" processes: exchange of force carrier particles. The "source code" of these processes is accessible in terms of Feynman diagram and its cousins in quantum field theory, which visually represent possible interaction scenarios of quantum particles. A "quantum amplitude" (a quantity that squares to the probability) is assigned to each diagram. The amplitude for propagating a particle (i.e., sending it from one location to another) is called a "propagator," while the amplitude for a local interaction is called a "vertex factor." Amplitudes are nothing but $(^0_n)$-tensors: they take $n$ input of incoming particle data and give a single complex number as output. For instance, a propagator is a $(^0_2)$-tensor, $\mathcal{M}_2(\texttt{data1},\texttt{data2})$. You may wonder, "why do we need two data when we are propagating only one particle?" This is because we understand the propagation of one particle as a process "two incoming particles collide and then annihilate." Of course, the propagator will be nonzero only if the momenta of the two particles sum to zero.

ㅤㅤI will ignore all the intricacies in field theory and continue pursuing this ("on-shell") particle point of view. Then, Wigner tells us that particles in flat spacetime are classified as follows.

• Massless particles: (momentum $p$, helicity $s$)
• Massive particles: (momentum $p$, spin $j$)

In other words, these are the "data type" that we will use in our programming of quantum interactions. The point is, the degrees of freedom carried by a particle depend on its "type." This is encoded in the numerator of the propagator, namely the "polarization sum."

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Propagator Structure

ㅤㅤFor example, let us consider the propagator $\mathcal{M}_2(\texttt{data1},\texttt{data2})$ for a massless particle. According to Wigner, it is $D\big(\{p_1,s_1\},\{p_2,s_2\}\big)$, where $\{p_1,s_1\}$ are the momentum and helicity data at one side and $\{p_2,s_2\}$ the other side. A typical formula is like this: \begin{align} \mathcal{M}_2\big(\{p_1,s_1\},\{p_2,s_2\}\big) = \frac{N(s_1,s_2)}{{p_1}^2+m^2}\kern1pt \delta^{(4)}(p_1+p_2) \,. \end{align} $m$ denotes the mass of the particle. It is conventional to drop off the "trivial" momentum delta function and just write \begin{align} D(s_1,s_2;p) = \frac{N(s_1,s_2)}{p^2+m^2} \,. \end{align}

ㅤㅤWe can convert this to the familiar textbook "Feynman-ish" expression with the aid of polarization vectors. For instance, a photon is a massless particle that has helicity $s=\pm1$. Similar to $p_1+p_2=0$, the only non-vanishing components of $N(s_1,s_2)$ are $N(+1,-1)=N(-1,+1)=1$. Therefore, \begin{align} D({}_\mu,{}_\nu;p) = \frac{\sum_{s_1,s_2} N(s_1,s_2)\kern1pt e^{s_1}{}_\mu e^{s_2}{}_\nu}{p^2+m^2} = \frac{e^+{}_\mu e^-{}_\nu+ e^-{}_\mu e^+{}_\nu}{p^2} = \frac{\eta_{\mu\nu} + 4p_{(\mu}\eta_{\nu)}}{p^2} \,. \end{align} $e^{\pm}{}_\mu$ are the polarization vectors, while the last line follows from the "XZC formalism" (1-2) that employs an auxiliary vector $\eta_\mu$ to retain Lorentz covariance. This makes contact with the use of auxiliary spinor in the spinor-helicity formalism (3), but you can just ignore it for the moment. Next, a graviton has helicity $s=\pm2$, and hence the propagator \begin{align} D({}_{\mu\nu},{}_{\rho\sigma};p) &= \frac{{e^+{}_\mu e^+{}_\nu} {e^-{}_\rho e^-{}_\sigma} + {e^-{}_\mu e^-{}_\nu} {e^+{}_\rho e^+{}_\sigma}}{p^2} \,,\\ &= \frac{\frac{1}{2}(\eta_{\mu\rho}\eta_{\nu\sigma} + \eta_{\mu\sigma}\eta_{\nu\rho}) - \frac{1}{2} \eta_{\mu\nu}\eta_{\rho\sigma} + (\text{auxiliary vector terms})}{p^2} \,. \end{align} The last line follows from... my Mathematica calculation! In sum, we rediscovered the photon propagator of Maxwell electromagnetism and the "trace reverser" propagator of linearized general relativity by considering only the physical particle degrees of freedom, namely the on-shell way of thinking.

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Classical Forces from 4pt Amplitudes

ㅤㅤFinally, the diagram describing the classical force is as the following.

Two particles with incident momenta $p_{1\mu}$ and $p_{2\mu}$ get scattered off by exchanging a force carrier particle (denoted by a wiggly line) carrying momentum $q_\mu$. We assume that the force carrier particle is massless. Now here's the rule. First, obtain the "quantum amplitude" $\mathcal{M}_4(p_1,p_2;-p_1-q,-p_2+q)$ by translating the above diagram into equations. Then, the Fourier transform \begin{align} \mathcal{I}(p_1,p_2;b) := -\int \frac{d^4q}{(2\pi)^4}\,(2\pi)\delta(2q\cdot p_1)\,(2\pi)\delta(2q\cdot p_2)\, e^{iq\cdot b}\, \mathcal{M}_4(p_1,p_2;-p_1-q,-p_2+q)|_{q^2\to0} \end{align} boils down to the classical potential between two particles separated by a large distance (impact parameter) $b^\mu$. Note that small $q^2$ corresponds to large $b^\mu$. This follows from the Born approximation. See also (4) and references therein.

ㅤㅤThe matter-photon coupling vertex is simply the Fourier-transformed electric current, $\tilde{J}^\mu(q)$. Therefore, \begin{align} \mathcal{M}_4 = \tilde{J}^\mu(-q) \,\frac{\eta_{\mu\nu}+\mathcal{O}(q^1)}{q^2}\, \tilde{J}^\nu(q) \,. \end{align} For scalar matter, it is $\tilde{J}^\mu(q) = e (p_1-p_2)^\mu = 2ep^\mu + \mathcal{O}(q^1)$ with $e$ being the coupling constant to electromagnetism, i.e., the electric charge. ($p_1$ and $p_2$ denote the incoming momenta for scalar legs 1 and 2.) Assuming classical sources being at rest to each other, we get \begin{align} - (2e_1 M_1 u^\mu)\, \frac{\eta_{\mu\nu}}{\vec{q}^2} \,(2e_2 M_2 u^\nu) = (2M_1)(2M_2)\cdot\left(\frac{e_1e_2}{\vec{q}^2}\right) \,, \end{align} which is the Fourier transform of the Coulomb potential $e_1e_2/4\pi |\vec{b}|$, peeling off the kinematic factors $(2M_1)(2M_2)$ (they cancel with the denominators of the Lorentz-invariant momentum space measure). $u^\mu$ is the four-velocity of each source, and we have imposed the delta functions $\delta(2q\cdot p_1)$ and $\delta(2q\cdot p_2)$ in its rest frame $u^\mu=(1,0,0,0)$. We see that the classical force mediated by a photon is repulsive for $e_1>0$, $e_2>0$.

ㅤㅤIn the case of gravity, the matter-graviton coupling vertex is $\kappa$ times the Fourier-transformed stress-energy tensor, $\kappa p^\mu p^\nu$. The universal gravitational coupling $\kappa$ is fixed as $\kappa = \sqrt{32\pi G}$ from the linearized Einstein-Hilbert action, demanding the canonical normalization of the graviton propagator. Then we get \begin{align} \mathcal{M}_4 = (\kappa {M_1}^2 u^\mu u^\nu) \,\frac{\frac{1}{2}(\eta_{\mu\rho}\eta_{\nu\sigma} + \eta_{\mu\sigma}\eta_{\nu\rho}) - \frac{1}{2} \eta_{\mu\nu}\eta_{\rho\sigma}+\mathcal{O}(q^1)}{q^2}\, (\kappa {M_2}^2 u^\rho u^\sigma) \,, \end{align} which boils down to \begin{align} - \kappa^2{M_1}^2{M_2}^2 \frac{(\textstyle (-1)(-1)-\frac{1}{2}(-1)(-1))}{\vec{q}^2} = (2M_1)(2M_2)\cdot \left(-\frac{4\pi G M_1 M_2}{\vec{q}^2}\right) \,. \end{align} This is the familiar Newtonian potential, attractive for $M_1>0$, $M_2>0$! Note that you get an attractive potential regardless of what value you assume for $\kappa$. Just for concreteness, we have referred to the Einstein-Hilbert action to fix $\kappa=\sqrt{32\pi G}$. But, what I want to emphasize is that the attractive nature of gravity can be derived without a Lagrangian description, using only the fact that its on-shell degrees of freedom are helicity $\pm2$ and the "well-established result" $1-\frac{1}{2}>0$. "Trumpets, please!"

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References

(1) Z. Xu, D.-H. Zhang and L. Chang, Tsinghua University preprints TUTP-84/3 (1984), TUTP-84/4 (1985); TUTP-84/5a (1985); Nucl. Phys. B291 (1987) 392.

(2) Dreiner, H. K., Haber, H. E., & Martin, S. P. (2010). Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry. Physics Reports, 494(1-2), 1-196.

(3) Elvang, H., & Huang, Y. T. (2013). Scattering amplitudes. arXiv preprint arXiv:1308.1697.

(4) Arkani-Hamed, N., Huang, Y. T., & O’Connell, D. (2020). Kerr black holes as elementary particles. Journal of High Energy Physics, 2020(1), 1-12.

Answer 2

The math behind spin of fields can be better understood using 2-spinor formalism and Twistor theory. In 3+1 dim space-times, given a 4-vector $v^a$, we can associate a Hermitian matrix $V^{AA'}$ defined by $V^{AA'}=v^a\sigma_a=\begin{bmatrix} V^0+V^3 & V^1+iV^2\\ V^1-iV^2 & V^0-V^3 \end{bmatrix}$. Using this representation, we can show that every orthochronous Lorentz transformation of $v^a$ corresponds to two spin transformations of $V^{AA'}$ , one being negative of the other: $V^{AA'}\to {t^A}_BV^{BB'}{\bar{t}^{A'}}_{B'}$, where ${t^A}_B$ matrices are elements of $SL(2,C)$. This is related to the fact that $SL(2,C)$ forms a double-cover of the orthochronous Lorentz group. Now we can write $V^{AA'}=\alpha^A\bar{\alpha}^{A'}$, where $\alpha^A$ and $\bar{\alpha}^{A'}$ are complex 2dimensional vectors and are known as spinors. An isomorphism exist between tangent vector space and the spin space: $T_pM\cong_{iso} S\otimes S'$. So $v^a\in T_pM$, $\alpha^A\in S^A$ and $\bar{\alpha}^{A'}\in S^{A'}$. The isomorphism is defined by the Infeld van der Warden symbols. In our example, the hermitian matrix $V^{AA'}=v^a\sigma_a= \alpha^A\bar{\alpha}^{A'}$, so the Infeld van der Warden symbols ${\sigma_a}^{AA'}$ are nothing but the Pauli matrices $\sigma^a$ (defined upto some normalization).

Using 2-spinor formalism we can express a skew symmetric tensor $F_{ab}$ (such as the Maxwell's fields) in terms of symmetric 2-spinor $\phi_{AB}$ : $F_{ab}=\phi_{AB}\epsilon_{A'B'}+c.c.$. (The epsilon matrix $\epsilon_{AB}$ is skew symmetric and is used to raise and lower spinor indices). Then the free Maxwell's equation $\nabla^aF_{ab}=0$ is equivalent to $\nabla^{AA'}\phi_{AB}=0$. Similarly, the vacuum Einstein's equation $\nabla^aC_{abcd}=0$ is equivalent to $\nabla^{AA'}\Psi_{ABCD}=0$.

A general spin-n/2 free massless field equation is given by: $$\nabla^{AA'}\phi_{ABC\cdots L}=0$$where $\phi_{ABCD}$ is symmetric in all n spinor indices $A,B,\cdots, L$

In classical relativistic dynamics, we can define heicity of massless particles using Pauli-Lubanski spin vector $W^a$. If we know the 4-momentum and 4-angular momentum of massless particle in COM frame, then the consistency of trajectory equation $P^aM_{ab}=0$ demands $W^a=sP^a$, where $s$ is the helicity. In a previous post Spin without quantum mechanics? , I have tried to explain the connection b/w helicity $s$ and spin-n/2 of massless fields $\phi_{ABC\cdots L}$, if we look at the representation of solution of the spinor field $\phi_{ABC\cdots L}$ in Twistor space (https://ncatlab.org/nlab/show/twistor+space). First we note that the solution for massless spinor field $\phi$ in Twistor space is given by Penrose-transform: $$\phi_{ABC\cdots L}=\frac{1}{2\pi i}\int_K \alpha_A\alpha_B\cdots \alpha_Lf(\alpha_R, -i\alpha_Rx^{RR'})\alpha_Md\alpha^M$$ The correct relativistic transformation of spinor field $\phi$ demands that the twistor function $f(\alpha_R, -i\alpha_Rx^{RR'})=f(Z)$ should have a homogeneity of degree $-n-2$ in twistor coordinates $Z_{\alpha}=(\alpha_A, -i\alpha_Ax^{AA'})$. We can also write the helicity $s$ of massless particle in terms of twistor coordinates: $s=\frac{1}{2}Z^{\alpha}\bar{Z}_{\alpha}$. Now if we consider quantization of Twistor space (i.e. we elevate the twistor coordinates $Z,\bar{Z}$ to twistor operators $\hat{Z}$ and $\hat{\bar{Z}}=-\frac{\partial}{\partial Z}$ satisfying the commutation relation $[Z^{\alpha},Z^{\beta}]=[\bar{Z}_{\alpha},\bar{Z}_{\beta}]=0$ and $[Z^{\alpha},\bar{Z}_{\beta}]=\delta^{\alpha}_{\beta}$), then we can identify spin operator $s\to \hat{S}:=\frac{1}{4}(Z^{\alpha}\bar{Z}_{\alpha}+\bar{Z}_{\alpha}Z^{\alpha})=\frac{1}{2}(Z^{\alpha}\bar{Z}_{\alpha}-2)$. Fact that twistor function $f(Z)$ is homogeneous of degree $-n-2$, we can write $Z^{\alpha}\bar{Z}_{\alpha}f(Z)=-Z^{\alpha}\frac{\partial}{\partial Z^{\alpha}}f(Z)=(n+2)f(Z)$. Thus $\hat{S}f(Z)=sf(Z)=\frac{n}{2}f(Z)$. This shows that after quantization of twistor space, the spin n/2 field $\phi$ has a helicity defined by $s=n/2$.

Thus for scalar field $\phi$ satisfying $\nabla^a\nabla_a\phi=0$, we have $n=0$ or helicity $s=0$. For Maxwell's field equation $\nabla^{AA'}\phi_{AB}=0$, $n=2$, so helicity $s=n/2=1$. Similarly for gravitational field $\Psi_{ABCD}$, $n=4$ or $s=2$ and so on..