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Defective Construction Work Under the Donoghue principle, if a builder carelessly gets the concrete mix wrong then the cost of putting right the defective concrete and other parasitic damage to the structure when it manifests itself later on is a purely economic cost, or loss. It is a cost that can be recovered through the contract, if there is one, and provided the contractual limitation period has not expired. But if there isn t a contract, or if the contractual limitation period has expired, then the Donoghue principle doesn t offer a route to a remedy for a purchaser against his builder or professional consultants because the loss is purely economic and outside the scope of the remedy. Nor does the Donoghue principle help a third party who has suffered purely economic loss, such as a subsequent purchaser or lessee, as the builder does not owe them a duty to avoid defects in the structure that result in the need to incur expenditure to effect repairs (see Department of Environment v. Thomas Bates & Sons Limited (1990)).



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with the notation z = q+i2p , Q = a+ia , and P = a ia The operator X is anti-self2 2i adjoint in H and is the in nitesimal generator of the unitary operator: z e X = e is e i( pQ qP ) = e is e za a = e is D (z)

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z The operator-valued map z D (z) = e za a = e i( pQ qP ) that appears in (347) is a unitary representation, up to a (crucial!) phase factor, of the group of translations of the complex plane Indeed, let us apply to the product D (z 1 )D (z 2 ) = e A1 e A2 , where Ai = z i a z i a and [A1 , A2 ] = 2i I(z 1 z 2 ) = 2iz 1 z 2 , the Weyl formula, 1 e A e B = e ( 2 [A,B ]) e (A+B ) , which is valid for any pair of operators that commute with their commutator, [A, [A, B]] = 0 = [B, [A, B]] One gets the composition rule

D (z 1 )D (z 2 ) = e iz 1 z 2 D (z 1 + z 2 )

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The unitarity of the operator D (z), D (z) 1 = D (z) = e z a za = D ( z) is then easy to check:

D (z)D ( z) = e iz z D (z z) = I d More generally, D (z n )D (z n 1) D (z 1 ) = e i D (z 1 + z 2 + + z n )

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The phase = j<k z j z k that appears in this expression has a topological (or symplectic) meaning: it is equal to the oriented area of the polygon A delimited by the path with vertices z 1 , z 1 + z 2 , , z 1 + z 2 + + z n , as shown in Figure 32 1 In canonical coordinates z i = 2 (q i + p i ), =

Just to complicate things a bit more, this next example adds an OrderBy clause:

The operators X, through the map X e X = e is D (z), thus de ne an irreducible unitary representation of the Weyl Heisenberg algebra w All other representations, except an unimportant case, are of the same type: there is uniqueness of the realization of the canonical commutation rules (Stone von Neumann theorem [21]) The phase factor e is that appears here commutes with all other operators in the representation but nevertheless plays a crucial role in quantum mechanics, since it encodes its noncommutativity, as will be seen below The D- function (it is actually an operator, also named displacement operator) D (z) = exp (za z a) is essential in determining the properties and applications of the coherent states Using the Weyl formula, one can also write D (z) = e

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z With this formulation and from a|0 = 0 e a |0 = |0 , one gets a new de nition of the standard coherent states, precisely the one that emerges from group theory [10, 11]:

You can use the printf command to help format the output so it looks better. First, let s just convert the print command to a printf command and see what that does:

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