Spin-2 Particle

[Thomas Merino]

Ascalar, a number, is unchanged (invariant) under a coordinate rotation so think of this as a "spin 0" object. Indeed, when we quantize a scalar field, the quanta have zero angular momentum; the quanta are spin 0 particles.

A vector, however, is covariant under a coordinate rotation. Importantly, if the coordinate system is rotated "once around", the vector is unchanged so think of this as a "spin 1" object; the vector rotates at the same rate as the coordinate system. More technically, to transform a vector, apply the transformation once.

Now, consider a rank 2 tensor (think of an outer product of two vectors as an example). To transform this object, the coordinate transformation must be applied twice (both vectors get the transformation).

When the coordinate system is rotated through half a rotation, the rank 2 tensor is unchanged so think of this object as a "spin 2" object; the rank 2 tensor rotates twice the rate of the coordinate system.

Spin-2 means that the spin is equal to 2 in the same sense in which spin-1 means that the spin is equal to 1 or spin-1/2 means that the spin is equal to 1/2. So it's hard to believe that you could understand the words spin-1/2 and spin-1 but not spin-2. It's like knowing how to drink half a liter of water, one liter of water, but be unable to drink 2 liters of water. Well, in this water example, it's actually more plausible.

The spin $\vec J$ is the intrinsic angular momentum. The intrinsic means "innate", the part of the angular momentum that exists even when the particle is at rest. The angular momentum is the quantity that is conserved whenever the laws of physics obey the rotational symmetry. The classical rotating object has $\vec J = \sum_i \vec r_i \times \vec p_i$ summed over the mass points.

I say harder because it's not really proper to talk of spinning faster or slower as, so far as elementary particles are pointlike objects one cannot give an extended geometry to them which is necessary to make sense of the moment of inertia that would let you figure out the angular velocity via

Hence we have to understand the spin solely through its angular momentum alone, without being able to attach a rotating reference frame to it unlike the case of, say, a spinning planet. The reason I say it as "harder" is because of an analogy of angular momentum to linear momentum: namely that linear momentum is "how much" motion is going on, under the idea that its elementary Newtonian form

can be thought of as "there's more motion if there's more stuff (larger $m$) and there's more motion if that stuff is moving faster (larger $v$)", as it intuitively "kind of feels so" that if there is such a quantity as "how much motion is happening", then having both twice the stuff moving and the same stuff moving twice as fast should both produce twice as much "motion going on" as a baseline scenario with a known "amount of motion going on". Of course, the impressive part is that this rather intuitive quantity happens to be conserved in Newtonian mechanics, which then is what lets us generalize the quantity to its relativistic and quantum counterparts.

Hence angular momentum is the same way, though $I$ is no longer clearly understandable as "amount of stuff", so the relevant concept we must transfer is just the idea it measures the "amount of rotational motion". An object with 2 units of momentum thus doesn't necessarily move twice as fast as one with 1 unit of momentum, but it is moving "more" in some way, so thus we could say perhaps, searching our vocabulary, that it is moving twice as "hard". Hence we could say an object with two units of angular momentum likewise spins twice as "hard" as one with one unit of angular momentum.

Is it correct if I say, when particles with spin are placed in a magnetic field they presess. The precession depends on the strength of the magnetic field.If the particles spin(revolve) on their own axes in a magnetic field, the gyration or the precession produced per rotation of the particle is termed as spin. That is to say, Spin 1 means the particle gyrates once every revolution. Spin 0 means there is no precession when the particle rotates. Spin 1/2 means every two revolutions of the particle produce one precession and so on. In other words spin is related with precession.

This is impossible to animate for a Boson which according to string theory as a Boson must be a closed loop manifold. The only way this hypothesized not yet discovered quantum spin-2 to be real would be IMHO what exactly in a previous answer here @The_Sympathizer mentioned "A spin-2 particle just spins "harder" than a spin-1 particle".

However, for massless Boson the theorized spin-2 graviton is theorized to be, this would be equivalent to a superluminal $v>c$ revolution speed spin-1 photon and impossible according to special relativity x2 "time contraction"! Instead of time dilation in the lab frame.

The earliest models for electron spin imagined a rotating charged mass, but this model fails when examined in detail: the required space distribution does not match limits on the electron radius: the required rotation speed exceeds the speed of light.[4] In the Standard Model, the fundamental particles are all considered "point-like": they have their effects through the field that surrounds them.[5] Any model for spin based on mass rotation would need to be consistent with that model.

Wolfgang Pauli, a central figure in the history of quantum spin, initially rejected any idea that the "degree of freedom" he introduced to explain experimental observations was related to rotation. He called it "classically non-describable two-valuedness". Later he allowed that it is related to angular momentum, but insisted on considering spin an abstract property.[6] This approach allowed Pauli to develop a proof of his fundamental Pauli exclusion principle, a proof now called the spin-statistics theorem.[7] In retrospect this insistence and the style of his proof initiated the modern particle physics era, where abstract quantum properties derived from symmetry properties dominate. Concrete interpretation became secondary and optional.[6]

Unlike classical wavefield circulation which allows continuous values of angular momentum, quantum wavefields allow only discrete values.[10] Consequently energy transfer to or from spins states always occurs in fixed quantum steps. Only a few steps are allowed: for many qualitative purposes the complexity of the spin quantum wavefields can be ignored and the system properties can be discussed in terms of "integer" or "half-integer" spin models as discussed in quantum numbers below.

Since elementary particles are point-like, self-rotation is not well-defined for them. However, spin implies that the phase of the particle depends on the angle as e i S θ \displaystyle e^iS\theta , for rotation of angle θ around the axis parallel to the spin S. This is equivalent to the quantum-mechanical interpretation of momentum as phase dependence in the position, and of orbital angular momentum as phase dependence in the angular position.

For fermions, the picture is less clear. Angular velocity is equal by the Ehrenfest theorem to the derivative of the Hamiltonian to its conjugate momentum, which is the total angular momentum operator J = L + S. Therefore, if the Hamiltonian H is dependent upon the spin S, dH/dS is non-zero, and the spin causes angular velocity, and hence actual rotation, i.e. a change in the phase-angle relation over time. However, whether this holds for free electron is ambiguous, since for an electron, S2 is constant, and therefore it is a matter of interpretation whether the Hamiltonian includes such a term. Nevertheless, spin appears in the Dirac equation, and thus the relativistic Hamiltonian of the electron, treated as a Dirac field, can be interpreted as including a dependence in the spin S.[9]

Composite particles also possess magnetic moments associated with their spin. In particular, the neutron possesses a non-zero magnetic moment despite being electrically neutral. This fact was an early indication that the neutron is not an elementary particle. In fact, it is made up of quarks, which are electrically charged particles. The magnetic moment of the neutron comes from the spins of the individual quarks and their orbital motions.

In ordinary materials, the magnetic dipole moments of individual atoms produce magnetic fields that cancel one another, because each dipole points in a random direction, with the overall average being very near zero. Ferromagnetic materials below their Curie temperature, however, exhibit magnetic domains in which the atomic dipole moments spontaneously align locally, producing a macroscopic, non-zero magnetic field from the domain. These are the ordinary "magnets" with which we are all familiar.

In paramagnetic materials, the magnetic dipole moments of individual atoms will partially align with an externally applied magnetic field. In diamagnetic materials, on the other hand, the magnetic dipole moments of individual atoms align oppositely to any externally applied magnetic field, even if it requires energy to do so.

The study of the behavior of such "spin models" is a thriving area of research in condensed matter physics. For instance, the Ising model describes spins (dipoles) that have only two possible states, up and down, whereas in the Heisenberg model the spin vector is allowed to point in any direction. These models have many interesting properties, which have led to interesting results in the theory of phase transitions.

In classical mechanics, the angular momentum of a particle possesses not only a magnitude (how fast the body is rotating), but also a direction (either up or down on the axis of rotation of the particle). Quantum-mechanical spin also contains information about direction, but in a more subtle form. Quantum mechanics states that the component of angular momentum for a spin-s particle measured along any direction can only take on the values[22]

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