Gravity and spin 2 representation

In summary, the spin-2 representation in gravity is quantized through a conserved, traceless and symmetric rank-2 tensor field. In the case of a non-zero mass, this tensor field has 9 components and requires 4 conditions to reduce it to 5 components. In the case of a massless field, the tensor field has 10 components and is reduced to 2 independent components through gauge transformations. Therefore, the massless spin-2 representation is carried by a symmetric rank-2 tensor with only 2 independent components.
  • #1
jk22
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I'm not at all involved in QG but from far away I noticed :

Spin 2 representations are 5x5 matrices.

But in gravity what mathematical objects are quantized ? If it's the metric then it's a 4x4 matrix so that cannot be that.

Or : how does quantization reveal a 5x5 matrix ?
 
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  • #2
jk22 said:
Spin 2 representations are 5x5 matrices.
This is true if, and only if, the object which carries the spin-2 representation has non-zero mass. In this case, the (massive) irreducible spin-2 representation (of the Poincare group) can be represented by a conserved, traceless and symmetric rank-2 tensor field: Consider the symmetric tensor field (i.e., we have 10 components) [itex]h^{\mu\nu} = h^{\nu\mu}[/itex], take its trace [itex]h = \eta_{\rho \sigma}h^{\rho\sigma}[/itex] then form the following traceless symmetric tensor field [tex]G^{\mu\nu} = h^{\mu\nu} - \frac{1}{4} \eta^{\mu\nu} h .[/tex] Now [itex]G^{\mu\nu}[/itex] has [itex]10 - 1= 9[/itex] components because [itex]G^{\mu\nu} = G^{\nu\mu}[/itex] and [itex]\eta_{\mu\nu}G^{\mu\nu} = 0[/itex]. So, to reduce the number of components to 5, we need to impose 4 more conditions: Usually, in field theory [itex]G^{\mu\nu}[/itex] is generated by a conserved source [itex]T^{\mu\nu} \ , \partial_{\mu}T^{\mu\nu} = 0[/itex], and satisfies the second-order equation [tex]( \partial^{2} + m^{2} ) G^{\mu\nu} = \alpha T^{\mu\nu} \ .[/tex] Therefore, the required 4 conditions on [itex]G^{\mu\nu}[/itex] are given by [itex]\partial_{\mu}G^{\mu\nu} = 0[/itex]. So, when a tensor field [itex]G^{\mu\nu}[/itex] satisfies the conditions,[itex]G^{\mu\nu} = G^{\nu\mu} \ , \eta_{\mu\nu}G^{\mu\nu} = 0[/itex] and [itex]\partial_{\mu}G^{\mu\nu} = 0[/itex], we say that [itex]G^{\mu\nu}[/itex] carries a massive irreducible spin-2 representation of the Poincare group.

In the massless case, we can show that the appropriate tensor is given by [tex]G^{\mu\nu} = h^{\mu\nu} - \frac{1}{2} \eta^{\mu\nu} h \ .[/tex] Notice, in this case, that [itex]G^{\mu\nu}[/itex] is still symmetric, but not traceless. So this tensor has 10 components. Again, we require [itex]G^{\mu\nu}[/itex] to be identically conserved, i.e., [itex]\partial_{\mu}G^{\mu\nu} = 0[/itex] (because, in this case the equation of motion has the form [itex]\partial^{2}G^{\mu\nu} = \beta T^{\mu\nu}[/itex]). Therefore, the number of components of [itex]G^{\mu\nu}[/itex] has been reduced to [itex]10 - 4 = 6[/itex]. However, we can show that theory is invariant under the following “gauge” transformations [tex]h^{\mu\nu} \to h^{\mu\nu} - \partial^{\mu} \chi^{\nu} - \partial^{\nu} \chi^{\mu} \ , [/tex]where [itex]\chi^{\mu}[/itex] is an arbitrary 4-vector field. This allows us to fix 4 out of the above 6 components of [itex]G^{\mu\nu}[/itex]. Thus, there are 2 and only 2 independent components left in [itex]G^{\mu\nu}[/itex] as it should be for massless fields. So, the massless “spin”-2 irreducible representation of the Poincare group is carried by a symmetric rank-2 tensor with only 2 independent components.
 
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1. What is the concept of gravity?

Gravity is a fundamental force of attraction that exists between all objects with mass. It is responsible for keeping planets in orbit around the sun, and objects on Earth from floating away into space.

2. How is gravity related to spin 2 representation?

In the framework of quantum field theory, gravity is represented by a spin 2 field. This means that gravity is described by particles that have a spin of 2, which is a type of intrinsic angular momentum.

3. What is spin 2 representation?

Spin 2 representation is a mathematical concept used to describe the properties and behavior of particles with a spin of 2. It is an important concept in quantum field theory and is used to study the behavior of fundamental forces, such as gravity.

4. How does spin 2 representation help us understand gravity?

Spin 2 representation allows us to mathematically describe the properties of gravity as a force. It helps us understand how gravity interacts with other particles and how it affects the curvature of spacetime.

5. Are there any other forces that are described by spin 2 representation?

Yes, in addition to gravity, the other fundamental force of nature, the weak nuclear force, is also described by spin 2 representation. This allows us to understand the similarities and differences between these two forces and how they interact with matter.

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