I want to say “I have no past” to express the following thought:
I don’t have a past, I only have the future, and I am to start anew".
Is “Ich habe keine Vergangenheit” correct?
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Well, yes, the translation is correct.
Note, however, that the concept itself is not a popular one - you can never completely leave your past behind, be it individual or collective. Attempting to do so will be seen as a weakness by many, as a way of attempting to shun responsibility or running away from something horrible instead of confronting it.
Saying something like that will immediately make everyone assume you are hiding a terrible secret, but I guess that's true for the English version as well.
The notion of starting anew is very well transported by the phrase
seine Vergangenheit hinter sich lassen – to leave one’s past behind
(Or if you want to formulate it as a resolution: “Ich will meine Vergangenheit hinter mir lassen.”)
Incidentally, Hulk used this construction in his answer. Your translation is also fine, it depends on the context in which you use it.
Ich lasse meine Vergangenheit hinter mir.
is maybe more appropriate if you want to forget terrible things that happend to you.
Ich habe keine Vergangenheit.
would be appropriate, e.g., for a criminal that wants to hide/forget the things he did.
The best thing however is to learn from the past and do better next time:
Ich will aus meinen Fehlern / meiner Vergangenheit lernen.
Gestern can be used in the sense of "past" in German. For example, someone who lives in the past would be jemand, der im Gestern lebt. So parallel to that:
Ich kenne kein Gestern (mehr).
I don't know a past (anymore) in the sense of I don't know you (anymore).
Imho it conveyes what you want to express more precisely than Ich habe keine Vergangenheit, although if you append mehr here, too, the aspect of "letting go" becomes more distinct, whereas the construction without it could be interpreted as "I did nothing with my life until now".
(Counterpart: Ich kenne nur (noch) ein Morgen.)
Yes, it's correct in that there is no more apposite word to express that specific concept. (There's Vorleben, but that's usually said by others about you, and with a negative connotation.)
It's somewhat unfortunate that this word is so long; compared to the dramatic and catchy "I have no past", the German rendition sounds considerably less stylish, but that sometimes happens. To approach the flavor of the original more closely you would have to reformulate quite a bit.