Defining Group Multiplication in Particle Physics

  • Thread starter quantumfireball
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In summary, In particle physics, three Lie groups (U(1), SU(2), and SU(3)) are commonly used. The multiplication of two groups of different dimensions, such as SU(2) X U(1) or SU(3) X SU(2) X U(1), is defined as the direct product or direct sum of the groups. This means that the set of all pairs of elements from each group is given a group structure, with the binary operation defined as the composition of the individual elements. This notation is often denoted as \bigoplus_{i=1}^n G_i, where G_i represents the individual groups.
  • #1
quantumfireball
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Everyone must be familiar with U(1),SU(2) and SU(3) Lie groups in particle physics .
But how does one define the multiplication of two groups of different dimensions
aka SU(2) X U(1) or SU(3) X SU(2) X U(1).
 
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  • #2
You aren't multiplying them. That is the direct product of the groups. Give two groups the direct product GxH is the set of all pairs (g,h) g in G, h in H with the natural composition

(g,h)(g',h')=(gg',hh')
 
  • #3
It's also the direct sum. If ever you encounter the notation [tex]\bigoplus_{i=1}^n G_i[/tex], this is what it means. Take the cartesian product of the G_i and give them a group structure by defining the binary operation as in matt grime's post. (g1,...,gn)(g1',...,gn'1)=(g1g1',...,gngn')
 
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1. What is the definition of multiplication?

Multiplication is a mathematical operation that involves combining two groups of objects to find the total number of objects in the new group.

2. How do you multiply two groups of numbers?

To multiply two groups of numbers, you can use the traditional method of long multiplication or use a calculator. For example, to multiply 3 groups of 4 objects each, you would write 3 x 4 = 12 or use a calculator to find the product of 3 and 4, which is also 12.

3. Can you multiply two groups with different numbers of objects?

Yes, you can multiply two groups with different numbers of objects. The resulting group will have a total number of objects equal to the product of the two groups. For example, if you multiply 2 groups of 3 and 4 objects, the resulting group will have a total of 12 objects.

4. What is the difference between multiplying two groups and adding them?

Multiplying two groups is a way to find the total number of objects in a new combined group, while adding two groups simply combines the two groups without finding the total number of objects. For example, if you have 2 groups of 3 objects each, multiplying them would give you a total of 6 objects in the new group, while adding them would simply give you 2 groups of 3 objects each.

5. Can you multiply two groups of fractions?

Yes, you can multiply two groups of fractions. To do so, you would multiply the numerators of the fractions together and then multiply the denominators together to get the final product. For example, if you have 2/3 of a group and 3/4 of another group, multiplying them would result in (2/3) x (3/4) = 6/12 = 1/2 of the total combined group.

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