As written in the discussion, this problem is there what ever the coefficients are. In your case they give 0=0. If you change them, they don't give this equality, and still it doesn't solve. Try this:

c=3*10^8
T=2
h=1.0545718* 10^(-34)
u={1,0,0,T}/Norm[{1,0,0,T}]
v={0,1,0,T}/Norm[{0,1,0,T}]
a={0,1,0,T}*{0,0,c,T}
b={1,1,1,T} *{c,c,c,T}
d= {1,0,1,T} *{c,c,c,T}

    pde =I*h^2*(u*a-u*b)*D[psi[x, y,z,t],x] -
I*h^2*(u*a)*D[psi[x, y,z,t],y]+
I*h^2*(u*b)* D[psi[x, y,z,t],z]-I*h^2*(v*d)*D[psi[x, y,z,t],z]+
I*h^2*(v*d)*(D[psi[x, y,z,t],y]) == 0

    dsol=DSolve[pde,{psiT[x,y,z,t],psiX[x,y,z,t],psiY[x,y,z,t],psiZ[x,y,z,t],
psiT[x,y,z,t],psiX[x,y,z,t],psiY[x,y,z,t],psiZ[x,y,z,t]},{t,x,y,z}];

DSolve", "deqx", "\"Supplied equations are not differential or integral equations of the given functions.\

So this 0=0 is not the problem. Something else is.

Hi Rohit, I found some inconsistencies with cdot and asterisk symbols, after correcting these, I get a working code, but a new and quite different problem:

c=3*10^8
T=2
h=1.0545718* 10^(-34)
u={1,0,0,T}/Norm[{1,0,0,T}]
v={0,1,0,T}/Norm[{0,1,0,T}]
a={0,1,0,T}*{0,0,c,T}
b={0,0,1,T} *{0,c,0,T}
d= {0,0,1,T} *{c,0,0,T}
pde =I*h^2*(u*a-u*b)*D[psi[x, y,z,t],x] -
I*h^2*(u*a)*D[psi[x, y,z,t],y]+I*h^2*(u*b)* D[psi[x, y,z,t],z]-
I*h^2*(v*d)*D[psi[x, y,z,t],z]+I*h^2*(v*d)*(D[psi[x, y,z,t],y]) == 0

Then when solving it:

dsol=DSolve[pde,{psiT[x,y,z,t],psiX[x,y,z,t],psiY[x,y,z,t],psiZ[x,y,z,t],psiT[x,y,z,t],psiX[x,y,z,t],psiY[x,y,z,t],psiZ[x,y,z,t]},{t,x,y,z}];

Supplied equations are not differential or integral equations of the given functions.

This happens no matter what values the vectors attain.

This should not be the case, if the vectors don't cancel out.