05a - Lecture 05 Define X Y and Z(all operators on C ∞ R...

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Unformatted text preview: Lecture 05 Define X , Y , and Z (all operators on C ∞ ( R , C )) by ( Xf )( t ) = tf ( t ) ( Y f )( t ) =- if ( t ) ( Zf )( t ) =- if ( t ) . It is easy to check that [ X,Y ] = Z , [ Z,X ] = 0, and [ Z,Y ] = 0. Since X , Y , and Z satisfy the same commutation relations as x , y , and z in h (the Heisenberg algebra), we get (by linear extension) a representation π of h on C ∞ ( R , C ) by π ( x ) = X , π ( y ) = Y , and π ( z ) = Z . Theorem 1. The space P comprising all polynomial functions R → C is a π-stable subspace of C ∞ ( R , C ) ; the induced representation of h on P is irreducible. (In terms of modules, P is a simple submodule of C ∞ ( R , C ) .) Proof. To see that P is simple, let Q ⊂ P be a nonzero submodule and pick a nonzero q ∈ Q , say q ( t ) = q + q 1 t + ··· + q n t n , where q n 6 = 0. Then Q 3 ( iY ) n q = q ( n ) = n ! q n . So, Q 3 1 n ! q n X m ( iY ) n = ‘ t m ’ for all m , i.e. Q = P . Definition 1. Define the annihilator a and the creator c by a = 1 √ 2 ( X + iY ) and c = 1 √ 2 ( X- iY ) ; also define A = √ 2 a and C = √ 2 c ....
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05a - Lecture 05 Define X Y and Z(all operators on C ∞ R...

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