What is Quotient? Understanding Algebraic Structures

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In summary, quotient spaces, groups, rings, fields, and vector spaces are all algebraic structures that are defined similarly with an equivalence relation. In the context of Quantum Mechanics, a Hilbert space is used to describe physical states, but there is too much information in the space. To make it bijective, the space is factorized by an equivalence relation, resulting in the projective Hilbert space. Normal subgroups are important in physics because they control the symmetries of things and are involved in group operations and representations.
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As one can see, the definition of quotient space, group, ring, field, vector space are very similar. It is similarly defined as an algebraic structure with a ~ on it. I am really having trouble vistualise what a quotient space and group are. My professor told me that we can work more easily with a quotient space since they are divided into mutual disjoint set. Is there another reason why it is defined in such a way?

Thx

Leon
 
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Lemme give you a physical example. In Quantum Mechanics one uses a (complex, separable) Hilbert space to describe physical (pure) quantum states. A Hilbert space is made up of vectors. There are physical reasons for which the mapping

{vectors}------>{physical (pure) states} is not one-to-one, i.e. injective. Worse, it maps an uncountable infinite set of vectors to one physical state.

So there's too much information in the Hilbert space (too many vectors) that we don't need. The application mentioned above is onto (surjective). To make it bijective, one factorizes the Hilbert space by this equivalence relation:

{vector A} ~ {vector B} iff the physical (pure) quantum state mapped from A is identical to the physical (pure) quantum state mapped from B.

In this way the bijective mapping is accomplished between the equivalence classes of vectors from the Hilbert space and the set of all physical (pure) quantum states.

The space H|~ is called "the projective Hilbert space" and provides the mathematical description of physical (pure) quantum states in Quantum Mechanics.

Daniel.

NOTE: The words in paranthesis: pure and separable are irrelevant for the point I'm trying to make. Just as the other words in paranthesis they're written for completeness.
 
  • #3
What about normal subgroup? How does it important in physics?
 
  • #4
All kernels are normal. Whenever groups are involved in anything, and maps of groups, you need to know about normal subgroups.

Groups control the symmetries of things. Some physical theories are described as the things that are left invariant by some group operation (one can characterize Galilean/Newtonian and relativistic mechanics this way). The group SU(3)xSU(2)xSU(1) is important in physics - representations of SU(2) correspond to spin states, for instance. A represnetation is a group hom into a vector space, so the kernel, being a normal subgroup is important...
 

Related to What is Quotient? Understanding Algebraic Structures

1. What is a quotient in algebra?

A quotient in algebra is the result of dividing two numbers or algebraic expressions. It is represented using the division symbol (/) and is read as "a divided by b".

2. How is a quotient different from a product in algebra?

A product in algebra is the result of multiplying two numbers or algebraic expressions, while a quotient is the result of dividing them. In other words, a product is the opposite of a quotient.

3. Can a quotient be a fraction?

Yes, a quotient can be a fraction. This is because when dividing two numbers or algebraic expressions, the result may not always be a whole number and can be represented as a fraction.

4. What are the properties of a quotient in algebra?

The properties of a quotient in algebra include the commutative property (a / b = b / a), the associative property ((a / b) / c = a / (b / c)), and the distributive property (a / (b + c) = (a / b) + (a / c)). These properties are used to manipulate and simplify algebraic expressions.

5. How is understanding quotient important in higher-level algebra?

Understanding quotient is important in higher-level algebra because it is used to solve more complex equations and problems. It is also a fundamental concept in other areas of mathematics, such as calculus and statistics.

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