There are some mistaken views regarding this famous chapter. The Psalmist speaks ( in our English translations ) with some words which throw a number of folks into confusion.
The elect are being referenced throughout. Words such as "our", "we", "we all", "us all", "many",and "their" refer to the elect . Think about it. Sheep are being discussed. In the New Testament the sheep are only His elect ones. It particularly comes out in John 10. In verse 11 of that chapter in John it says that the good shepherd lays down his life for the sheep. In verse 15 Jesus says that : I lay down my life for the sheep . Verse 28 says : I give them eternal life.
Everyone does not receive eternal life -- only the ones He died for -- laid down His life for -- His sheep. The Lamb died for His sheep. The wrath of God came down on Jesus for them . He did not die for goats ( see Matt.25). He died for the ones he knew, not for the ones to whom He will say : I never knew you. The Lord knows those who are His (2 Tim. 2:19 ). He knows His sheep (John 10:27). His Father gave them (His sheep) to Jesus.
In 1 Peter 2:25 Peter writes that : For you were like sheep going astray, but now you have returned to the Shepherd and Overseer of your souls.
In Isaiah 53:1 the answer to the question is that the arm of the Lord has been revealed to the elect alone.
In verse 4 : Surely he took up OUR infirmities. Whose infirmities ? Ours -- not the infirmities of each and every person past, present and future. A specific group is targeted. He bore the sins of His people.
Verse 5 says : But he was pierced for OUR transgressions, he was crushed for OUR iniquities; the punishment that brought US peace was upon him, and by his wounds We are healed.
Our, Our, Us, We -- the elect .
Verse 6 : WE all , like sheep ... the Lord has laid on him the iniquity of Us all. That means the sins of all whom He was to save.
Verse 8 tells us that : for the trangression of my people he was stricken.
His people are the sheep referenced throughout this chapter.
In verse 11 it states that he will justify many, and He will bear THEIR iniquities. Matthew Henry relates : "Thus He shall bear the iniquities of the many that He designed to justify."
Within verse 12 it says that : he bore the sin of many, and made intercession for the transgressors.
The "many" (not all people who ever were and shall be) He gave His life for -- the sin which he bore for them . IOW -- as many as He gave His life a ransom for.
In Ephesians 5:2 it says that Christ loved US and gave Himself for US. Verse 25 goes on to say that : Christ loved the Church and gave Himself up for Her. The Church and the sheep are one and the same.
Jesus made satisfaction for the sins of His people -- the same ones who shall not recieve any charge against them -- the elect ( Rom. 8:33 ). He was made sin for US. WE are the ones for whom He died. He was OUR Substitute. We sing : "in our place condemned He stood ... " He stood in the place of, on behalf of, in the room of, for --- His sheep .