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captain
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what is the tensor product's physical significance? I know what it does mathematically, but what does it mean. I have looked on textbooks and wikipedia but i still can't understand the physical signifcance.
The physical signifigance will depend on the particular application.captain said:what is the tensor product's physical significance? I know what it does mathematically, but what does it mean. I have looked on textbooks and wikipedia but i still can't understand the physical signifcance.
Reverie said:As was mentioned previously, the physical significance depends on the application. Maybe this explanation will help.
Let V be a three dimensional vector space with basis {e1,e2,e3}, and let W be a four dimensional vector space with basis {f1,f2,f3,f4}.
Then V tensor W is a 12 dimensional vector space with basis
e1 tensor f1
e1 tensor f2
e1 tensor f3
e1 tensor f4
e2 tensor f1
e2 tensor f2
e2 tensor f3
e2 tensor f4
e3 tensor f1
e3 tensor f2
e3 tensor f3
e3 tensor f4
Furthermore, V direct sum W is a 7 dimensional vector space with basis:
(e1,0)
(e2,0)
(e3,0)
(0,f1)
(0,f2)
(0,f3)
(0,f4)
See the difference?
In Quantum Mechanics, spin is often considered. There is spin up and spin down. Sometimes the spin state is tensored with a wave function. Then, we have
wavefunction tensor spinstate=(wavefunction+,wavefunction-).
-Reverie
mathwonk said:fortunately there is no such thing as an,...ayeeeee!
The tensor product is a mathematical operation that combines two vector spaces to create a new vector space. In physics, it is used to describe the properties of physical systems that are composed of multiple subsystems. The resulting tensor represents the joint state of the subsystems and describes how they interact with each other.
In quantum mechanics, the tensor product is used to describe the composite state of multiple particles or systems. The resulting tensor represents the entangled state of the particles and describes how they are correlated and influenced by each other's measurements.
The tensor product is a multiplication operation between two vectors, while the tensor sum is an addition operation. The tensor product results in a larger vector space, while the tensor sum results in a smaller vector space. In physics, the tensor sum is used to describe the addition of physical quantities, while the tensor product is used to describe the combination of physical systems.
In relativity, tensor product is used to describe the properties of spacetime. The tensor product of two vectors in spacetime represents the joint state of two events and describes their relative position and momentum. It is also used to describe the curvature of spacetime in Einstein's field equations.
Tensor product has many applications in physics, engineering, and computer science. It is used in quantum computing, signal processing, image recognition, and machine learning. In physics, it is used in quantum field theory, general relativity, and fluid dynamics. It also has applications in chemistry, biology, and economics.